#set document(title: "2.2 Some Basic Functions", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 2.2#h(0.6em)Some Basic Functions In this section, we study the graphs of some important basic functions. Many functions fall into families or classes of similar functions, and recognizing the appropriate family for a given situation is an important part of modeling. We begin by reviewing the absolute value. === Absolute Value The absolute value is used to discuss problems involving distance. For example, consider the number line shown below. Starting at the origin, we travel in opposite directions to reach the two numbers #math.equation(block: false, alt: "6")[$6$] and #math.equation(block: false, alt: "minus 6")[$− 6$], but the #emph[distance] we travel in each case is the same. #figure(figph[number line with absolute value illustrated], alt: "number line with absolute value illustrated", caption: none) The distance from a number #math.equation(block: false, alt: "c")[$c$] to the origin is called the #strong[absolute value] of #math.equation(block: false, alt: "c")[$c$], denoted by #math.equation(block: false, alt: "vertical bar c vertical bar")[$| c |$]. Because distance is never negative, the absolute value of a number is always positive (or zero). Thus, #math.equation(block: false, alt: "vertical bar 6 vertical bar equals 6")[$| 6 | = 6$] and #math.equation(block: false, alt: "vertical bar minus 6 vertical bar equals 6")[$| − 6 | = 6$]. In general, we define the absolute value of a number #math.equation(block: false, alt: "x")[$x$] as follows. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Absolute Value] The absolute value of #math.equation(block: false, alt: "x")[$x$] is defined by #math.equation(block: true, alt: "vertical bar x vertical bar equals open brace x, if x greater than or equal to 0; minus x, if x less than 0")[$| x | = \{ x & "if " x ≥ 0 \ − x & "if " x < 0$] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ This definition says that the absolute value of a positive number (or zero) is the same as the number. To find the absolute value of a negative number, we take the opposite of the number, which results in a positive number. For instance, #math.equation(block: true, alt: "vertical bar minus 6 vertical bar equals minus open parenthesis minus 6 close parenthesis equals 6")[$| − 6 | = − ( − 6 ) = 6$] ] If #math.equation(block: false, alt: "vertical bar x vertical bar equals minus x")[$| x | = − x$], what can you say about #math.equation(block: false, alt: "x")[$x$]? \_\_\_\_\_ #solutionbox[ #math.equation(block: false, alt: "x")[$x$] must be zero or negative if #math.equation(block: false, alt: "vertical bar x vertical bar equals minus x")[$| x | = − x$]. ] If #math.equation(block: false, alt: "vertical bar x vertical bar equals minus x")[$| x | = − x$], what can you say about #math.equation(block: false, alt: "x")[$x$]? + #math.equation(block: false, alt: "x")[$x$] must be zero. + #math.equation(block: false, alt: "x")[$x$] must be negative. + #math.equation(block: false, alt: "x")[$x$] must be zero or negative. + This cannot happen for any value of #math.equation(block: false, alt: "x")[$x$]. Absolute value bars act like grouping devices in the order of operations: You should complete any operations that appear inside absolute value bars before you compute the absolute value. #examplebox("Example 1")[][ Simplify each expression. + #math.equation(block: false, alt: "vertical bar 3 minus 8 vertical bar")[$| 3 − 8 |$] + #math.equation(block: false, alt: "vertical bar 3 vertical bar minus vertical bar 8 vertical bar")[$| 3 | − | 8 |$] #solutionbox[ + We simplify the expression inside the absolute value bars first. #math.equation(block: true, alt: "vertical bar 3 minus 8 vertical bar equals vertical bar minus 5 vertical bar equals 5")[$| 3 − 8 | = | − 5 | = 5$] + We simplify each absolute value; then subtract. #math.equation(block: true, alt: "vertical bar 3 vertical bar minus vertical bar 8 vertical bar equals 3 minus 8 equals minus 5")[$| 3 | − | 8 | = 3 − 8 = − 5$] ] ] Simplify each expression. + #math.equation(block: false, alt: "12 minus 3 vertical bar minus 6 vertical bar equals")[$12 − 3 | − 6 | =$]\_\_\_\_\_ + #math.equation(block: false, alt: "minus 7 minus 3 vertical bar 2 minus 9 vertical bar equals")[$− 7 − 3 | 2 − 9 | =$]\_\_\_\_\_ #solutionbox[ + #math.equation(block: false, alt: "minus 6")[$− 6$] + #math.equation(block: false, alt: "minus 28")[$− 28$] ] Simplify each expression. + #math.equation(block: false, alt: "12 minus 3 vertical bar minus 6 vertical bar")[$12 − 3 | − 6 |$] + #math.equation(block: false, alt: "minus 7 minus 3 vertical bar 2 minus 9 vertical bar")[$− 7 − 3 | 2 − 9 |$] #solutionbox[ + #math.equation(block: false, alt: "minus 6")[$− 6$] + #math.equation(block: false, alt: "minus 28")[$− 28$] ] === Examples of Models Many situations can be modeled by a handful of simple functions. The following examples represent applications of eight useful functions. The contractor for a new hotel is estimating the cost of the marble tile for a circular lobby. The cost is a function of the #emph[square] of the diameter of the lobby. The number of board-feet that can be cut from a Ponderosa pine is a function of the #emph[cube] of the circumference of the tree at a standard height. #figure(figph[parabola], alt: "parabola", caption: none) #figure(figph[cubic], alt: "cubic", caption: none) The manager of an appliance store must decide how many coffee-makers to order every quarter. The optimal order size is a function of the #emph[square root] of the annual demand for coffeemakers. Investors are deciding whether to support a windmill farm. The wind speed needed to generate a given amount of power is a function of the #emph[cube root] of the power. #figure(figph[square root], alt: "square root", caption: none) #figure(figph[cube root], alt: "cube root", caption: none) The frequency of the note produced by a violin string is a function of the #emph[reciprocal] of the length of the string. The loudness, or intensity, of the music at a concert is a function of the #emph[reciprocal of the square] of your distance from the speakers. #figure(figph[inverse], alt: "inverse", caption: none) #figure(figph[cube root], alt: "cube root", caption: none) The annual return on an investment is a linear function of the interest rate. You are flying from Los Angeles to New York. Your distance from the Mississippi River is an #emph[absolute value] function of time. #figure(figph[linear], alt: "linear", caption: none) #figure(figph[absolute value], alt: "absolute value", caption: none) We will consider each of these functions and their applications in more detail in later sections. For now, you should become familiar with the properties of each graph and be able to sketch them easily from memory. #notebox("Investigation", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Eight Basic Functions] Part I Some Powers + Complete the table of values for the squaring function, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared")[$f ( x ) = x^(2)$], and the cubing function, #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x cubed")[$g ( x ) = x^(3)$]. Then sketch each function on graph paper, using the table values to help you scale the axes. + Verify both graphs with your graphing calculator. + State the intervals on which each graph is increasing. + Write a few sentences comparing the two graphs. The graph of #math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$] is called a #strong[parabola], and the graph of #math.equation(block: false, alt: "y equals x cubed")[$y = x^(3)$] is called a #strong[cubic]. #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared")[$f ( x ) = x^(2)$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x cubed")[$g ( x ) = x^(3)$]]), [#math.equation(block: false, alt: "minus 3")[$− 3$]], [], [], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [], [], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [], [], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [], [], [#math.equation(block: false, alt: "0")[$0$]], [], [], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [], [], [#math.equation(block: false, alt: "1")[$1$]], [], [], [#math.equation(block: false, alt: "2")[$2$]], [], [], [#math.equation(block: false, alt: "3")[$3$]], [], [], )) Part II Some Roots + Complete the tables for the square root function, #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x")[$f ( x ) = sqrt(x)$], and the cube root function, #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the cube root of x")[$g ( x ) = root(3, x)$]. (Round your answers to two decimal places.) Then sketch each function on graph paper, using the table values to help you scale the axes. + Verify both graphs with your graphing calculator. + State the intervals on which each graph is increasing. + Write a few sentences comparing the two graphs. #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x")[$f ( x ) = sqrt(x)$]]), [#math.equation(block: false, alt: "0")[$0$]], [], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [], [#math.equation(block: false, alt: "1")[$1$]], [], [#math.equation(block: false, alt: "2")[$2$]], [], [#math.equation(block: false, alt: "3")[$3$]], [], [#math.equation(block: false, alt: "4")[$4$]], [], [#math.equation(block: false, alt: "5")[$5$]], [], [#math.equation(block: false, alt: "7")[$7$]], [], [#math.equation(block: false, alt: "9")[$9$]], [], )) #figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the cube root of x")[$g ( x ) = root(3, x)$]]), [#math.equation(block: false, alt: "minus 8")[$− 8$]], [], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [], [#math.equation(block: false, alt: "0")[$0$]], [], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [], [#math.equation(block: false, alt: "1")[$1$]], [], [#math.equation(block: false, alt: "4")[$4$]], [], [#math.equation(block: false, alt: "8")[$8$]], [], )) Part III Asymptotes + Complete the table for the functions #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction 1 over x and g open parenthesis x close parenthesis equals the fraction 1 over x squared")[$f ( x ) = display(frac(1, x)) " " " and " " " g ( x ) = display(frac(1, x^(2)))$] What is true about #math.equation(block: false, alt: "f open parenthesis 0 close parenthesis")[$f ( 0 )$] and #math.equation(block: false, alt: "g open parenthesis 0 close parenthesis")[$g ( 0 )$]? + Prepare a grid on graph paper, scaling both axes from #math.equation(block: false, alt: "minus 5")[$− 5$] to #math.equation(block: false, alt: "5")[$5$]. Plot the points from the table and connect them with smooth curves. + As #math.equation(block: false, alt: "x")[$x$] increases through larger and larger values, what happens to the values of #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]? Extend your graph to reflect your answer. + What happens to #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] as #math.equation(block: false, alt: "x")[$x$] decreases through larger and larger negative values (that is, for #math.equation(block: false, alt: "x equals minus 5 , minus 6 , minus 7 , and so on")[$x = − 5 , − 6 , − 7 , …$])? Extend your graph for these #math.equation(block: false, alt: "x")[$x$]-values. #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 1 over x")[$f ( x ) = display(frac(1, x))$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 1 over x squared")[$g ( x ) = display(frac(1, x^(2)))$]]), [#math.equation(block: false, alt: "minus 4")[$− 4$]], [], [], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [], [], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [], [], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [], [], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [], [], [#math.equation(block: false, alt: "0")[$0$]], [], [], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [], [], [#math.equation(block: false, alt: "1")[$1$]], [], [], [#math.equation(block: false, alt: "2")[$2$]], [], [], [#math.equation(block: false, alt: "3")[$3$]], [], [], [#math.equation(block: false, alt: "4")[$4$]], [], [], )) As the values of #math.equation(block: false, alt: "x")[$x$] get larger in absolute value, the graph approaches the #math.equation(block: false, alt: "x")[$x$]-axis. However, because #math.equation(block: false, alt: "the fraction 1 over x")[$display(frac(1, x))$] never #emph[equals] zero for any #math.equation(block: false, alt: "x")[$x$]-value, the graph never actually touches the #math.equation(block: false, alt: "x")[$x$]-axis. We say that the #math.equation(block: false, alt: "x")[$x$]-axis is a #strong[horizontal asymptote] for the graph. Repeat step (3) for the graph of #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]. Next we'll examine the graphs of #math.equation(block: false, alt: "f")[$f$] and #math.equation(block: false, alt: "g")[$g$] near #math.equation(block: false, alt: "x equals 0")[$x = 0$]. + Use your calculator to evaluate #math.equation(block: false, alt: "f")[$f$] for several #math.equation(block: false, alt: "x")[$x$]-values close to zero and record the results in the tables below.#figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 1 over x")[$f ( x ) = display(frac(1, x))$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 1 over x squared")[$g ( x ) = display(frac(1, x^(2)))$]]), [#math.equation(block: false, alt: "minus 2")[$− 2$]], [], [], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [], [], [#math.equation(block: false, alt: "minus 0.1")[$− 0.1$]], [], [], [#math.equation(block: false, alt: "minus 0.01")[$− 0.01$]], [], [], [#math.equation(block: false, alt: "minus 0.001")[$− 0.001$]], [], [], )) #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 1 over x")[$f ( x ) = display(frac(1, x))$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 1 over x squared")[$g ( x ) = display(frac(1, x^(2)))$]]), [#math.equation(block: false, alt: "2")[$2$]], [], [], [#math.equation(block: false, alt: "1")[$1$]], [], [], [#math.equation(block: false, alt: "0.1")[$0.1$]], [], [], [#math.equation(block: false, alt: "0.01")[$0.01$]], [], [], [#math.equation(block: false, alt: "0.001")[$0.001$]], [], [], )) #linebreak() What happens to the values of #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] as #math.equation(block: false, alt: "x")[$x$] approaches zero? Extend your graph of #math.equation(block: false, alt: "f")[$f$] to reflect your answer. #linebreak() As #math.equation(block: false, alt: "x")[$x$] approaches zero from the left (through negative values), the function values decrease toward #math.equation(block: false, alt: "minus ∞")[$− upright(∞)$]. As #math.equation(block: false, alt: "x")[$x$] approaches zero from the right (through positive values), the function values increase toward #math.equation(block: false, alt: "∞")[$upright(∞)$]. The graph approaches but never touches the vertical line #math.equation(block: false, alt: "x equals 0")[$x = 0$] (the #math.equation(block: false, alt: "y")[$y$]-axis.)We say that the graph of #math.equation(block: false, alt: "f")[$f$] has a #strong[vertical asymptote] at #math.equation(block: false, alt: "x equals 0")[$x = 0$]. + Repeat step (1) for the graph of #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]. + The functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 1 over x")[$f ( x ) = display(frac(1, x))$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 1 over x squared")[$g ( x ) = display(frac(1, x^(2)))$] are examples of #strong[rational functions], so called because they are fractions, or ratios. Verify both graphs with your graphing calculator. Use the window #math.equation(block: true, alt: "Xmin, equals minus 4, Xmax equals 4; Ymin, equals minus 4, Ymax equals 4")[$"Xmin" & = − 4 & & "Xmax" = 4 \ "Ymin" & = − 4 & & "Ymax" = 4$] + State the intervals on which each graph is increasing. + Write a few sentences comparing the two graphs. Part IV Absolute Value + Complete the table for the two functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x")[$f ( x ) = x$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals vertical bar x vertical bar")[$g ( x ) = | x |$]. Then sketch each function on graph paper, using the table values to help you scale the axes. + Verify both graphs with your graphing calculator. Your calculator uses the notation #math.equation(block: false, alt: "a b s open parenthesis x close parenthesis")[$a b s ( x )$] instead of #math.equation(block: false, alt: "vertical bar x vertical bar")[$| x |$] for the absolute value of #math.equation(block: false, alt: "x")[$x$]. First, position the cursor after #math.equation(block: false, alt: "Y sub 1 equals")[$Y_(1) =$] in the graphing window. Now access the absolute value function by pressing 2nd #math.equation(block: false, alt: "0")[$0$] for #emph[CATALOG]; then ENTER for #emph[abs(]. Don’t forget to press #math.equation(block: false, alt: "X")[$X$] if you want to graph #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$]. + State the intervals on which each graph is increasing. + Write a few sentences comparing the two graphs. #figure(table( columns: 3, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x")[$f ( x ) = x$]], [#math.equation(block: false, alt: "g open parenthesis x close parenthesis equals vertical bar x vertical bar")[$g ( x ) = | x |$]]), [#math.equation(block: false, alt: "minus 4")[$− 4$]], [], [], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [], [], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [], [], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [], [], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [], [], [#math.equation(block: false, alt: "0")[$0$]], [], [], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [], [], [#math.equation(block: false, alt: "1")[$1$]], [], [], [#math.equation(block: false, alt: "2")[$2$]], [], [], [#math.equation(block: false, alt: "3")[$3$]], [], [], [#math.equation(block: false, alt: "4")[$4$]], [], [], )) ] === Graphs of Eight Basic Functions The graphs of the eight basic functions considered in Eight Basic Functions are shown below. Once you know the shape of each graph, you can sketch an accurate picture by plotting a few guidepoints and drawing the curve through those points. Usually, points (or vertical asymptotes!) at #math.equation(block: false, alt: "x equals minus 1")[$x = − 1$], #math.equation(block: false, alt: "0")[$0$], and #math.equation(block: false, alt: "1")[$1$] make good guidepoints. #figure(figph[parabola, cubic, square root], alt: "parabola, cubic, square root", caption: none) #figure(figph[cube root, reciprocal, inverse square], alt: "cube root, reciprocal, inverse square", caption: none) #figure(figph[line, absolute value], alt: "line, absolute value", caption: none) Which of the eight basic functions are undefined at #math.equation(block: false, alt: "x equals 0")[$x = 0$]? Which are undefined for negative #math.equation(block: false, alt: "x")[$x$]? Which are always positive? \_\_\_\_\_ Which of the eight basic functions are undefined at #math.equation(block: false, alt: "x equals 0")[$x = 0$]? Which are undefined for negative #math.equation(block: false, alt: "x")[$x$]? Which are always positive? === Some Properties of Absolute Value In Functions, we saw that for most functions, #math.equation(block: false, alt: "f open parenthesis a plus b close parenthesis")[$f ( a + b )$] is not equal to #math.equation(block: false, alt: "f open parenthesis a close parenthesis plus f open parenthesis b close parenthesis")[$f ( a ) + f ( b )$]. We may be able to find #emph[some] values of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] for which #math.equation(block: false, alt: "f open parenthesis a plus b close parenthesis equals f open parenthesis a close parenthesis plus f open parenthesis b close parenthesis")[$f ( a + b ) = f ( a ) + f ( b )$] is true, but if it is not true for #emph[all] values of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$], we cannot claim that #math.equation(block: false, alt: "f open parenthesis a plus b close parenthesis equals f open parenthesis a close parenthesis plus f open parenthesis b close parenthesis")[$f ( a + b ) = f ( a ) + f ( b )$] for that function. For example, for the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared")[$f ( x ) = x^(2)$], if we choose #math.equation(block: false, alt: "a equals 3")[$a = 3$] and #math.equation(block: false, alt: "b equals 4")[$b = 4$], then #math.equation(block: true, alt: "f open parenthesis 3 plus 4 close parenthesis, equals f open parenthesis 7 close parenthesis equals 7 squared equals 49; but f open parenthesis 3 close parenthesis plus f open parenthesis 4 close parenthesis, equals 3 squared plus 4 squared equals 9 plus 16 equals 25")[$f ( 3 + 4 ) & = f ( 7 ) = 7^(2) = 49 \ "but " " " " " " " " " " " f ( 3 ) + f ( 4 ) & = 3^(2) + 4^(2) = 9 + 16 = 25$] so we have proved that #math.equation(block: false, alt: "f open parenthesis a plus b close parenthesis not equal to f open parenthesis a close parenthesis plus f open parenthesis b close parenthesis")[$f ( a + b ) ≠ f ( a ) + f ( b )$] for the squaring function. (In fact, we already knew this because #math.equation(block: false, alt: "open parenthesis a plus b close parenthesis squared not equal to a squared plus b squared")[$( a + b )^(2) ≠ a^(2) + b^(2)$] as long as neither #math.equation(block: false, alt: "a")[$a$] nor #math.equation(block: false, alt: "b")[$b$] is #math.equation(block: false, alt: "0")[$0$].) What about multiplication? Which of the basic functions have the property that #math.equation(block: false, alt: "f open parenthesis a b close parenthesis equals f open parenthesis a close parenthesis f open parenthesis b close parenthesis")[$f ( a b ) = f ( a ) f ( b )$] for all #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]? You will consider this question in the homework problems, but in particular you will need to recall the following properties of absolute value. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Properties of Absolute Value] #math.equation(block: true, alt: "vertical bar a plus b vertical bar less than or equal to vertical bar a vertical bar plus vertical bar b vertical bar, Triangle inequality; vertical bar a b vertical bar equals vertical bar a vertical bar vertical bar b vertical bar, Multiplicative property")[$| a + b | ≤ | a | + | b | & & "Triangle inequality" \ | a b | = | a | | b | & & "Multiplicative property "$] ] #examplebox("Example 2")[][ Verify the triangle inequality for three cases: #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] are both positive, #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] are both negative, and #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] have opposite signs. #solutionbox[ - We choose positive values for #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$], say #math.equation(block: false, alt: "a equals 3")[$a = 3$] and #math.equation(block: false, alt: "b equals 5")[$b = 5$]. Then #math.equation(block: true, alt: "vertical bar 3 plus 5 vertical bar equals vertical bar 8 vertical bar equals 8 and vertical bar 3 vertical bar plus vertical bar 5 vertical bar equals 3 plus 5 equals 8")[$| 3 + 5 | = | 8 | = 8 " " " and " " " | 3 | + | 5 | = 3 + 5 = 8$] so #math.equation(block: false, alt: "vertical bar 3 plus 5 vertical bar equals vertical bar 3 vertical bar plus vertical bar 5 vertical bar")[$| 3 + 5 | = | 3 | + | 5 |$]. - For the second case, we choose #math.equation(block: false, alt: "a equals minus 3")[$a = − 3$] and #math.equation(block: false, alt: "b equals minus 5")[$b = − 5$]. Then #math.equation(block: true, alt: "vertical bar minus 3 plus open parenthesis minus 5 close parenthesis vertical bar equals vertical bar minus 8 vertical bar equals 8 and vertical bar minus 3 vertical bar plus vertical bar minus 5 vertical bar equals 3 plus 5 equals 8")[$| − 3 + ( − 5 ) | = | − 8 | = 8 " " " and " " " | − 3 | + | − 5 | = 3 + 5 = 8$] so #math.equation(block: false, alt: "vertical bar minus 3 plus open parenthesis minus 5 close parenthesis vertical bar equals vertical bar minus 3 vertical bar plus vertical bar minus 5 vertical bar")[$| − 3 + ( − 5 ) | = | − 3 | + | − 5 |$]. - For the third case, we choose #math.equation(block: false, alt: "a equals 3")[$a = 3$] and #math.equation(block: false, alt: "b equals minus 5")[$b = − 5$]. Then #math.equation(block: true, alt: "vertical bar 3 plus open parenthesis minus 5 close parenthesis vertical bar equals vertical bar minus 2 vertical bar equals 2 and vertical bar 3 vertical bar plus vertical bar minus 5 vertical bar equals 3 plus 5 equals 8")[$| 3 + ( − 5 ) | = | − 2 | = 2 " " " and " " " | 3 | + | − 5 | = 3 + 5 = 8$] so #math.equation(block: false, alt: "vertical bar 3 plus open parenthesis minus 5 close parenthesis vertical bar less than vertical bar 3 vertical bar plus vertical bar minus 5 vertical bar")[$| 3 + ( − 5 ) | < | 3 | + | − 5 |$]. In each case, #math.equation(block: false, alt: "vertical bar a plus b vertical bar less than or equal to vertical bar a vertical bar plus vertical bar b vertical bar")[$| a + b | ≤ | a | + | b |$]. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Note that #emph[verifying] a statement for one or two values of the variables does not #emph[prove] the statement is true for #emph[all] values of the variables. However, working with examples can help us understand the meaning and significance of mathematical properties. ] Verify the multiplicative property of absolute value for the three cases in the previous example. - Case #math.equation(block: false, alt: "a equals 3")[$a = 3$] and #math.equation(block: false, alt: "b equals 5")[$b = 5$] #linebreak() Does #math.equation(block: false, alt: "vertical bar 3 vertical bar vertical bar 5 vertical bar equals vertical bar 3 times 5 vertical bar")[$| 3 | | 5 | = | 3 ⋅ 5 |$]? \_\_\_\_\_ - Case #math.equation(block: false, alt: "a equals minus 3")[$a = − 3$] and #math.equation(block: false, alt: "b equals minus 5")[$b = − 5$] #linebreak() Does #math.equation(block: false, alt: "vertical bar minus 3 vertical bar vertical bar minus 5 vertical bar equals vertical bar open parenthesis minus 3 close parenthesis times open parenthesis minus 5 close parenthesis vertical bar")[$| − 3 | | − 5 | = | ( − 3 ) ⋅ ( − 5 ) |$]? \_\_\_\_\_ - Case #math.equation(block: false, alt: "a equals 3")[$a = 3$] and #math.equation(block: false, alt: "b equals minus 5")[$b = − 5$] #linebreak() Does #math.equation(block: false, alt: "vertical bar 3 vertical bar vertical bar minus 5 vertical bar equals vertical bar 3 open parenthesis minus 5 close parenthesis vertical bar")[$| 3 | | − 5 | = | 3 ( − 5 ) |$]? \_\_\_\_\_ #solutionbox[ - #math.equation(block: false, alt: "vertical bar 3 vertical bar vertical bar 5 vertical bar equals 15 equals vertical bar 3 times 5 vertical bar")[$| 3 | | 5 | = 15 = | 3 ⋅ 5 |$] - #math.equation(block: false, alt: "vertical bar minus 3 vertical bar vertical bar minus 5 vertical bar equals 15 equals vertical bar open parenthesis minus 3 close parenthesis times open parenthesis minus 5 close parenthesis vertical bar")[$| − 3 | | − 5 | = 15 = | ( − 3 ) ⋅ ( − 5 ) |$] - #math.equation(block: false, alt: "vertical bar 3 vertical bar vertical bar minus 5 vertical bar equals 15 equals vertical bar 3 open parenthesis minus 5 close parenthesis vertical bar")[$| 3 | | − 5 | = 15 = | 3 ( − 5 ) |$] ] Verify the multiplicative property of absolute value for the three cases in the previous example. - Case #math.equation(block: false, alt: "a equals 3")[$a = 3$] and #math.equation(block: false, alt: "b equals 5")[$b = 5$] #linebreak() Does #math.equation(block: false, alt: "vertical bar 3 vertical bar vertical bar 5 vertical bar equals vertical bar 3 times 5 vertical bar")[$| 3 | | 5 | = | 3 ⋅ 5 |$]? - Case #math.equation(block: false, alt: "a equals minus 3")[$a = − 3$] and #math.equation(block: false, alt: "b equals minus 5")[$b = − 5$] #linebreak() Does #math.equation(block: false, alt: "vertical bar minus 3 vertical bar vertical bar minus 5 vertical bar equals vertical bar open parenthesis minus 3 close parenthesis times open parenthesis minus 5 close parenthesis vertical bar")[$| − 3 | | − 5 | = | ( − 3 ) ⋅ ( − 5 ) |$]? - Case #math.equation(block: false, alt: "a equals 3")[$a = 3$] and #math.equation(block: false, alt: "b equals minus 5")[$b = − 5$] #linebreak() Does #math.equation(block: false, alt: "vertical bar 3 vertical bar vertical bar minus 5 vertical bar equals vertical bar 3 open parenthesis minus 5 close parenthesis vertical bar")[$| 3 | | − 5 | = | 3 ( − 5 ) |$]? #solutionbox[ - #math.equation(block: false, alt: "vertical bar 3 vertical bar vertical bar 5 vertical bar equals 15 equals vertical bar 3 times 5 vertical bar")[$| 3 | | 5 | = 15 = | 3 ⋅ 5 |$] - #math.equation(block: false, alt: "vertical bar minus 3 vertical bar vertical bar minus 5 vertical bar equals 15 equals vertical bar open parenthesis minus 3 close parenthesis times open parenthesis minus 5 close parenthesis vertical bar")[$| − 3 | | − 5 | = 15 = | ( − 3 ) ⋅ ( − 5 ) |$] - #math.equation(block: false, alt: "vertical bar 3 vertical bar vertical bar minus 5 vertical bar equals 15 equals vertical bar 3 open parenthesis minus 5 close parenthesis vertical bar")[$| 3 | | − 5 | = 15 = | 3 ( − 5 ) |$] ] Which of these is true for all #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]? \_\_\_\_\_ #solutionbox[ #math.equation(block: false, alt: "vertical bar a minus b vertical bar greater than or equal to vertical bar a vertical bar minus vertical bar b vertical bar")[$| a − b | ≥ | a | − | b |$] for all #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]. ] Which of these is true for all #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]? + #math.equation(block: false, alt: "vertical bar a minus b vertical bar less than or equal to vertical bar a vertical bar minus vertical bar b vertical bar")[$| a − b | ≤ | a | − | b |$] + #math.equation(block: false, alt: "vertical bar a minus b vertical bar greater than or equal to vertical bar a vertical bar minus vertical bar b vertical bar")[$| a − b | ≥ | a | − | b |$] + #math.equation(block: false, alt: "vertical bar a minus b vertical bar greater than or equal to vertical bar a vertical bar plus vertical bar b vertical bar")[$| a − b | ≥ | a | + | b |$] + #math.equation(block: false, alt: "vertical bar a plus b vertical bar greater than or equal to vertical bar a vertical bar plus vertical bar b vertical bar")[$| a + b | ≥ | a | + | b |$] === Functions Defined Piecewise A function may be defined by different formulas on different portions of the #math.equation(block: false, alt: "x")[$x$]-axis. Such a function is said to be defined #strong[piecewise]. To graph a function defined piecewise, we consider each piece of the #math.equation(block: false, alt: "x")[$x$]-axis separately. #examplebox("Example 3")[][ Graph the function defined by #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals open brace x plus 1, if x less than or equal to 1; 3, if x greater than 1")[$f ( x ) = \{ x + 1 & "if " x ≤ 1 \ 3 & "if " x > 1$] #solutionbox[ Think of the plane as divided into two regions by the vertical line #math.equation(block: false, alt: "x equals 1")[$x = 1$], as shown below. In the left-hand region (#math.equation(block: false, alt: "x less than or equal to 1")[$x ≤ 1$]), we graph the line #math.equation(block: false, alt: "y equals x plus 1")[$y = x + 1$]. (The fastest way to graph the line is to plot its intercepts, #math.equation(block: false, alt: "open parenthesis minus 1 , 0 close parenthesis")[$( − 1 , 0 )$] and #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$].) Notice that the value #math.equation(block: false, alt: "x equals 1")[$x = 1$] is included in the first region, so #math.equation(block: true, alt: "f open parenthesis 1 close parenthesis equals 1 plus 1 equals 2")[$f ( 1 ) = 1 + 1 = 2$] , and the point #math.equation(block: false, alt: "open parenthesis 1 , 2 close parenthesis")[$( 1 , 2 )$] is included on the graph. We indicate this with a solid dot at the point #math.equation(block: false, alt: "open parenthesis 1 , 2 close parenthesis")[$( 1 , 2 )$]. In the right-hand region (#math.equation(block: false, alt: "x greater than 1")[$x > 1$]), we graph the horizontal line #math.equation(block: false, alt: "y equals 3")[$y = 3$]. #figure(figph[piecewise function], alt: "piecewise function", caption: none) The value #math.equation(block: false, alt: "x equals 1")[$x = 1$] is not included in the second region, so the point #math.equation(block: false, alt: "open parenthesis 1 , 3 close parenthesis")[$( 1 , 3 )$] is not part of the graph. We indicate this with an open circle at the point #math.equation(block: false, alt: "open parenthesis 1 , 3 close parenthesis")[$( 1 , 3 )$]. ] ] Graph the piecewise defined function #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals open brace minus 1 minus x, if x less than or equal to minus 1; x cubed, if x greater than minus 1")[$g ( x ) = \{ − 1 − x & " if " x ≤ − 1 \ x^(3) & " if " x > − 1$] #solutionbox[ #figure(figph[piecewise defined function], alt: "piecewise defined function", caption: none) ] Graph the piecewise defined function #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals open brace minus 1 minus x, if x less than or equal to minus 1; x cubed, if x greater than minus 1")[$g ( x ) = \{ − 1 − x & " if " x ≤ − 1 \ x^(3) & " if " x > − 1$] #solutionbox[ #figure(figph[piecewise defined function], alt: "piecewise defined function", caption: none) ] In Practice 3, what is the value of #math.equation(block: false, alt: "g open parenthesis minus 1 close parenthesis")[$g ( − 1 )$]? \_\_\_\_\_ #solutionbox[ #math.equation(block: true, alt: "g open parenthesis minus 1 close parenthesis equals 0")[$g ( − 1 ) = 0$] ] In Practice 3, what is the value of #math.equation(block: false, alt: "g open parenthesis minus 1 close parenthesis")[$g ( − 1 )$]? + #math.equation(block: false, alt: "0")[$0$] + #math.equation(block: false, alt: "minus 1")[$− 1$] + #math.equation(block: false, alt: "minus 2")[$− 2$] + undefined The absolute value function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals vertical bar x vertical bar")[$f ( x ) = | x |$] is an example of a function that is defined piecewise. #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals vertical bar x vertical bar equals open brace x, if x greater than or equal to 0; minus x, if x less than 0")[$f ( x ) = | x | = \{ x & "if " x ≥ 0 \ − x & "if " x < 0$] To sketch the absolute value function, we graph the line #math.equation(block: false, alt: "y equals x")[$y = x$] in the first quadrant and the line #math.equation(block: false, alt: "y equals minus x")[$y = − x$] in the second quadrant. #examplebox("Example 4")[][ + Write a piecewise definition for #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals vertical bar x minus 3 vertical bar")[$g ( x ) = | x − 3 |$]. + Sketch a graph of #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals vertical bar x minus 3 vertical bar")[$g ( x ) = | x − 3 |$]. #solutionbox[ + In the definition for #math.equation(block: false, alt: "vertical bar x vertical bar")[$| x |$], we replace #math.equation(block: false, alt: "x")[$x$] by #math.equation(block: false, alt: "x minus 3")[$x − 3$] to get #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals vertical bar x minus 3 vertical bar equals open brace x minus 3, if x minus 3 greater than or equal to 0; minus open parenthesis x minus 3 close parenthesis, if x minus 3 less than 0")[$g ( x ) = | x − 3 | = \{ x − 3 & "if " " " x − 3 ≥ 0 \ − ( x − 3 ) & "if " " " x − 3 < 0$] We can simplify this expression to #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals vertical bar x minus 3 vertical bar equals open brace x minus 3, if x greater than or equal to 3; minus x plus 3, if x less than 3")[$g ( x ) = | x − 3 | = \{ x − 3 & "if " " " x ≥ 3 \ − x + 3 & "if " " " x < 3$] + In the first region, #math.equation(block: false, alt: "x greater than or equal to 3")[$x ≥ 3$], we graph the line #math.equation(block: false, alt: "y equals x minus 3")[$y = x − 3$]. Because #math.equation(block: false, alt: "x equals 3")[$x = 3$] is included in this region, the endpoint of this portion of the graph, #math.equation(block: false, alt: "open parenthesis 3 , 0 close parenthesis")[$( 3 , 0 )$], is included, too. #linebreak() In the second region, #math.equation(block: false, alt: "x less than 3")[$x < 3$], we graph the line #math.equation(block: false, alt: "y equals minus x plus 3")[$y = − x + 3$]. Note that the two pieces of the graph meet at the point #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$], as shown below.#figure(figph[piecewise function], alt: "piecewise function", caption: none) ] ] + Use your graphing utility to graph #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals vertical bar x minus 3 vertical bar")[$g ( x ) = | x − 3 |$] and #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals vertical bar x vertical bar plus vertical bar minus 3 vertical bar")[$h ( x ) = | x | + | − 3 |$]. Are the graphs the same? \_\_\_\_\_ + Explain why the functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals vertical bar x plus k vertical bar")[$f ( x ) = | x + k |$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals vertical bar x vertical bar plus vertical bar k vertical bar")[$g ( x ) = | x | + | k |$] are not the same if #math.equation(block: false, alt: "k not equal to 0")[$k ≠ 0$]. #linebreak() \_\_\_\_\_ #solutionbox[ + No, the graphs of #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals vertical bar x minus 3 vertical bar")[$g ( x ) = | x − 3 |$] and #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals vertical bar x vertical bar plus vertical bar minus 3 vertical bar")[$h ( x ) = | x | + | − 3 |$] are not the same. + Because #math.equation(block: false, alt: "vertical bar x plus k vertical bar not equal to vertical bar x vertical bar plus vertical bar k vertical bar")[$| x + k | ≠ | x | + | k |$] when #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "k")[$k$] have opposite signs. ] + Use your graphing utility to graph #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals vertical bar x minus 3 vertical bar")[$g ( x ) = | x − 3 |$] and #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals vertical bar x vertical bar plus vertical bar minus 3 vertical bar")[$h ( x ) = | x | + | − 3 |$]. Are the graphs the same? + Explain why the functions #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals vertical bar x plus k vertical bar")[$f ( x ) = | x + k |$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals vertical bar x vertical bar plus vertical bar k vertical bar")[$g ( x ) = | x | + | k |$] are not the same if #math.equation(block: false, alt: "k not equal to 0")[$k ≠ 0$]. #solutionbox[ + No, the graphs of #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals vertical bar x minus 3 vertical bar")[$g ( x ) = | x − 3 |$] and #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals vertical bar x vertical bar plus vertical bar minus 3 vertical bar")[$h ( x ) = | x | + | − 3 |$] are not the same. + Because #math.equation(block: false, alt: "vertical bar x plus k vertical bar not equal to vertical bar x vertical bar plus vertical bar k vertical bar")[$| x + k | ≠ | x | + | k |$] when #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "k")[$k$] have opposite signs. ] What relationship do you see between the graphs of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed")[$f ( x ) = x^(3)$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the cube root of x")[$g ( x ) = root(3, x)$]? Between the graphs of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared")[$f ( x ) = x^(2)$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the square root of x")[$g ( x ) = sqrt(x)$]? \_\_\_\_\_ What relationship do you see between the graphs of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed")[$f ( x ) = x^(3)$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the cube root of x")[$g ( x ) = root(3, x)$]? Between the graphs of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared")[$f ( x ) = x^(2)$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the square root of x")[$g ( x ) = sqrt(x)$]? === Section Summary ==== Vocabulary Look up the definitions of new terms in the Glossary. - Absolute value - Verify - Triangle inequality - Vertical asymptote - Rational function - Parabola - Piecewise defined function - Multiplicative property - Guidepoints - Horizontal asymptote - Cubic ==== CONCEPTS + The #strong[absolute value] of #math.equation(block: false, alt: "x")[$x$] is defined by #math.equation(block: true, alt: "vertical bar x vertical bar equals open brace x, if x greater than or equal to 0; minus x, if x less than 0")[$| x | = \{ x & "if " x ≥ 0 \ − x & "if " x < 0$] + The absolute value has the following properties:#figure(table( columns: 2, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "vertical bar a plus b vertical bar less than or equal to vertical bar a vertical bar plus vertical bar b vertical bar")[$| a + b | ≤ | a | + | b |$]], [Triangle inequality]), [#math.equation(block: false, alt: "vertical bar a b vertical bar equals vertical bar a vertical bar vertical bar b vertical bar")[$| a b | = | a | | b |$]], [Multiplicative property], )) + Many useful functions fall into families or classes of variations on basic functions. + We can make sketches of the eight basic functions using guidepoints. + Functions can be defined piecewise, with different formulas on different intervals. ==== STUDY QUESTIONS + Is it true that #math.equation(block: false, alt: "minus x")[$− x$] must be a negative number? Why or why not? + Are there any numbers for which #math.equation(block: false, alt: "x equals minus x")[$x = − x$]? + If #math.equation(block: false, alt: "0 less than x less than 1")[$0 < x < 1$], which is larger, #math.equation(block: false, alt: "x squared")[$x^(2)$] or #math.equation(block: false, alt: "x cubed")[$x^(3)$]? + If #math.equation(block: false, alt: "0 less than x less than 1")[$0 < x < 1$], which is larger, #math.equation(block: false, alt: "the square root of x")[$sqrt(x)$] or #math.equation(block: false, alt: "the cube root of x")[$root(3, x)$]? + List the eight basic functions considered in this section. + Which of the eight basic functions have a horizontal asymptote? A vertical asymptote? + What does an open circle on a graph mean? + For what value(s) of #math.equation(block: false, alt: "x")[$x$] does #math.equation(block: false, alt: "vertical bar x plus 6 vertical bar equals 0")[$| x + 6 | = 0$]? ==== SKILLS Practice each skill in the Homework problems listed. + Simplify expressions containing absolute values: \#1–10 + Sketch graphs of the basic functions by hand: \#15–18 + Identify the graph of a basic function: \#19–26 + Solve equations and inequalities graphically: \#11–14, 27–34 + Graph functions defined piecewise: \#41–58 === Homework 2.2 For problems 1–10, simplify the expression according to the order of operations. + #math.equation(block: false, alt: "minus vertical bar minus 9 vertical bar")[$− | − 9 |$] + #math.equation(block: false, alt: "minus open parenthesis minus 9 close parenthesis")[$− ( − 9 )$] + #math.equation(block: false, alt: "minus 9")[$− 9$] + #math.equation(block: false, alt: "9")[$9$] + #math.equation(block: false, alt: "2 minus open parenthesis minus 6 close parenthesis")[$2 − ( − 6 )$] + #math.equation(block: false, alt: "2 minus vertical bar minus 6 vertical bar")[$2 − | − 6 |$] + #math.equation(block: false, alt: "vertical bar minus 8 vertical bar minus vertical bar 12 vertical bar")[$| − 8 | − | 12 |$] + #math.equation(block: false, alt: "vertical bar minus 8 minus 12 vertical bar")[$| − 8 − 12 |$] + #math.equation(block: false, alt: "minus 4")[$− 4$] + #math.equation(block: false, alt: "20")[$20$] + #math.equation(block: false, alt: "vertical bar minus 3 vertical bar plus vertical bar minus 5 vertical bar")[$| − 3 | + | − 5 |$] + #math.equation(block: false, alt: "vertical bar minus 3 plus open parenthesis minus 5 close parenthesis vertical bar")[$| − 3 + ( − 5 ) |$] #math.equation(block: true, alt: "4 minus 9 vertical bar 2 minus 8 vertical bar")[$4 − 9 | 2 − 8 |$] #math.equation(block: true, alt: "minus 50")[$− 50$] #math.equation(block: true, alt: "2 minus 5 vertical bar minus 6 minus 3 vertical bar")[$2 − 5 | − 6 − 3 |$] #math.equation(block: true, alt: "vertical bar minus 4 minus 5 vertical bar vertical bar 1 minus 3 open parenthesis minus 5 close parenthesis vertical bar")[$| − 4 − 5 | | 1 − 3 ( − 5 ) |$] #math.equation(block: true, alt: "144")[$144$] #math.equation(block: true, alt: "vertical bar minus 3 plus 7 vertical bar vertical bar minus 2 open parenthesis 6 minus 10 close parenthesis vertical bar")[$| − 3 + 7 | | − 2 ( 6 − 10 ) |$] #math.equation(block: true, alt: "vertical bar vertical bar minus 5 vertical bar minus vertical bar minus 6 vertical bar vertical bar")[$| | − 5 | − | − 6 | |$] #math.equation(block: true, alt: "1")[$1$] #math.equation(block: true, alt: "vertical bar vertical bar 4 vertical bar minus vertical bar minus 6 vertical bar vertical bar")[$| | 4 | − | − 6 | |$] In Problems 11–14, show how to use the graphs to find the values. Estimate your answers to one decimal point. Compare your estimates to values obtained with a calculator. Refer to the graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed")[$f ( x ) = x^(3)$]. + Estimate the value of #math.equation(block: false, alt: "open parenthesis 1.4 close parenthesis cubed")[$( 1.4 )^(3)$]. + Find all numbers whose cubes are #math.equation(block: false, alt: "minus 20")[$− 20$]. + Find all solutions of the equation #math.equation(block: false, alt: "x cubed equals 6")[$x^(3) = 6$]. + Estimate the value of #math.equation(block: false, alt: "the cube root of 24")[$root(3, 24)$]. #figure(figph[cubic], alt: "cubic", caption: none) + #math.equation(block: false, alt: "2.7")[$2.7$] + #math.equation(block: false, alt: "minus 2.7")[$− 2.7$] + #math.equation(block: false, alt: "1.8")[$1.8$] + #math.equation(block: false, alt: "2.9")[$2.9$] Refer to the graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared")[$f ( x ) = x^(2)$]. + Estimate the value of #math.equation(block: false, alt: "open parenthesis minus 2.5 close parenthesis squared")[$( − 2.5 )^(2)$]. + Find all numbers whose squares are #math.equation(block: false, alt: "12")[$12$]. + Find all solutions of the equation #math.equation(block: false, alt: "x squared equals 15")[$x^(2) = 15$]. + Estimate the value of #math.equation(block: false, alt: "the square root of 10.5")[$sqrt(10.5)$]. #figure(figph[quadratic], alt: "quadratic", caption: none) Refer to the graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 1 over x")[$f ( x ) = display(frac(1, x))$]. #figure(figph[reciprocal], alt: "reciprocal", caption: none) + Estimate the value of #math.equation(block: false, alt: "the fraction 1 over 3.4")[$display(frac(1, 3.4))$]. + Find all numbers whose reciprocals are #math.equation(block: false, alt: "minus 2.5")[$− 2.5$]. + Find all solutions of the equation #math.equation(block: false, alt: "the fraction 1 over x equals 4.8")[$display(frac(1, x)) = 4.8$]. + #math.equation(block: false, alt: "0.3")[$0.3$] + #math.equation(block: false, alt: "minus 0.4")[$− 0.4$] + #math.equation(block: false, alt: "0.2")[$0.2$] Refer to the graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals vertical bar x minus 2 vertical bar")[$f ( x ) = | x − 2 |$]. #figure(figph[quadratic], alt: "quadratic", caption: none) + Estimate the value of #math.equation(block: false, alt: "vertical bar 1.6 minus 2 vertical bar")[$| 1.6 − 2 |$]. + Find all values of #math.equation(block: false, alt: "x")[$x$] for which #math.equation(block: false, alt: "vertical bar x minus 2 vertical bar equals 3")[$| x − 2 | = 3$]. + Find all solutions of the equation #math.equation(block: false, alt: "vertical bar x minus 2 vertical bar equals 0.4")[$| x − 2 | = 0.4$]. For Problems 15–18, + Sketch both functions on the same grid, paying attention to the shape of the graph. Plot at least three guidepoints for each graph to ensure accuracy. + Use the graph to find all solutions of the equation #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals g open parenthesis x close parenthesis")[$f ( x ) = g ( x )$]. + On what intervals is #math.equation(block: false, alt: "f open parenthesis x close parenthesis greater than g open parenthesis x close parenthesis")[$f ( x ) > g ( x )$]? #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x squared , g open parenthesis x close parenthesis equals x cubed")[$f ( x ) = x^(2) , " " g ( x ) = x^(3)$] + #figure(figph[quadratic and cubic], alt: "quadratic and cubic", caption: none) + #math.equation(block: false, alt: "x equals 0 , x equals 1")[$x = 0 , " " x = 1$] + #math.equation(block: false, alt: "open parenthesis minus ∞ , 0 close parenthesis")[$( − upright(∞) , 0 )$] and #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of x , g open parenthesis x close parenthesis equals the cube root of x")[$f ( x ) = sqrt(x) , " " g ( x ) = root(3, x)$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction 1 over x , g open parenthesis x close parenthesis equals the fraction 1 over x squared")[$f ( x ) = display(frac(1, x)) , " " g ( x ) = display(frac(1, x^(2)))$] + #figure(figph[quadratic and cubic], alt: "quadratic and cubic", caption: none) + #math.equation(block: false, alt: "x equals 1")[$x = 1$] + #math.equation(block: false, alt: "open parenthesis 1 , plus ∞ close parenthesis")[$( 1 , + upright(∞) )$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x , g open parenthesis x close parenthesis equals vertical bar x vertical bar")[$f ( x ) = x , " " g ( x ) = | x |$] For Problems 19–24, graph each set of functions together in an appropriate window. Describe how graphs (b) and (c) are different from the basic graph. + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x cubed")[$f ( x ) = x^(3)$] + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x cubed minus 2")[$g ( x ) = x^(3) − 2$] + #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals x cubed plus 1")[$h ( x ) = x^(3) + 1$] Graph (b) is the basic graph shifted 2 units down; graph (c) is the basic graph shifted 1 unit up. + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals vertical bar x vertical bar")[$f ( x ) = | x |$] + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals vertical bar x minus 2 vertical bar")[$g ( x ) = | x − 2 |$] + #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals vertical bar x plus 1 vertical bar")[$h ( x ) = | x + 1 |$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 1 over x")[$f ( x ) = display(frac(1, x))$] + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 1 over x plus 1.5")[$g ( x ) = display(frac(1, x + 1.5))$] + #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the fraction 1 over x minus 1")[$h ( x ) = display(frac(1, x − 1))$] Graph (b) is the basic graph shifted 1.5 units left; graph (c) is the basic graph shifted 1 unit right. + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 1 over x squared")[$f ( x ) = display(frac(1, x^(2)))$] + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 1 over x squared plus 2")[$g ( x ) = display(frac(1, x^(2))) + 2$] + #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the fraction 1 over x squared minus 1")[$h ( x ) = display(frac(1, x^(2))) − 1$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x")[$f ( x ) = sqrt(x)$] + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals minus the square root of x")[$g ( x ) = − sqrt(x)$] + #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the square root of minus x")[$h ( x ) = sqrt(− x)$] Graph (b) is the basic graph reflected about the #math.equation(block: false, alt: "x")[$x$]-axis; graph (c) is the basic graph reflected about the #math.equation(block: false, alt: "y")[$y$]-axis. + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the cube root of x")[$f ( x ) = root(3, x)$] + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals minus the cube root of x")[$g ( x ) = − root(3, x)$] + #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the cube root of minus x")[$h ( x ) = root(3, − x)$] Each graph in Problems 25–26 is a variation of one of the eight basic graphs of Eight Basic Functions. Identify the basic graph for each problem. #figure(figph[six GC graphs], alt: "six GC graphs", caption: none) + #math.equation(block: false, alt: "the square root of x")[$sqrt(x)$] + #math.equation(block: false, alt: "the cube root of x")[$root(3, x)$] + #math.equation(block: false, alt: "vertical bar x vertical bar")[$| x |$] + #math.equation(block: false, alt: "the fraction 1 over x")[$display(frac(1, x))$] + #math.equation(block: false, alt: "x cubed")[$x^(3)$] + #math.equation(block: false, alt: "the fraction 1 over x squared")[$display(frac(1, x^(2)))$] #figure(figph[six GC graphs], alt: "six GC graphs", caption: none) In Problems 27–30, use the graph to estimate the solution to the equation or inequality. Show the solution or solutions on the graph. Then check your answers algebraically. The figure shows a graph of #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x minus 2")[$f ( x ) = sqrt(x) − 2$], for #math.equation(block: false, alt: "x greater than 0")[$x > 0$]. Solve the following: #figure(figph[square root shifted], alt: "square root shifted", caption: none) + #math.equation(block: false, alt: "the square root of x minus 2 equals 1.5")[$sqrt(x) − 2 = 1.5$] + #math.equation(block: false, alt: "the square root of x minus 2 equals 2.25")[$sqrt(x) − 2 = 2.25$] + #math.equation(block: false, alt: "the square root of x minus 2 less than 1")[$sqrt(x) − 2 < 1$] + #math.equation(block: false, alt: "the square root of x minus 2 greater than minus 0.25")[$sqrt(x) − 2 > − 0.25$] + #math.equation(block: false, alt: "x approximately equals 12")[$x ≈ 12$] + #math.equation(block: false, alt: "x approximately equals 18")[$x ≈ 18$] + #math.equation(block: false, alt: "x less than 9")[$x < 9$] + #math.equation(block: false, alt: "x greater than 3")[$x > 3$] The figure shows a graph of #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 4 over x plus 2")[$g ( x ) = display(frac(4, x + 2))$], for #math.equation(block: false, alt: "x greater than minus 2")[$x > − 2$]. Solve the following: #figure(figph[reciprocal shifted], alt: "reciprocal shifted", caption: none) + #math.equation(block: false, alt: "the fraction 4 over x plus 2 equals 4")[$display(frac(4, x + 2)) = 4$] + #math.equation(block: false, alt: "the fraction 4 over x plus 2 equals 0.8")[$display(frac(4, x + 2)) = 0.8$] + #math.equation(block: false, alt: "the fraction 4 over x plus 2 greater than 1")[$display(frac(4, x + 2)) > 1$] + #math.equation(block: false, alt: "the fraction 4 over x plus 2 less than 3")[$display(frac(4, x + 2)) < 3$] The figure shows a graph of #math.equation(block: false, alt: "w open parenthesis t close parenthesis equals minus 10 open parenthesis t plus 1 close parenthesis cubed plus 10")[$w ( t ) = − 10 ( t + 1 )^(3) + 10$]. Solve the following: #figure(figph[cubic transformed], alt: "cubic transformed", caption: none) + #math.equation(block: false, alt: "minus 10 open parenthesis t plus 1 close parenthesis cubed plus 10 equals 100")[$− 10 ( t + 1 )^(3) + 10 = 100$] + #math.equation(block: false, alt: "minus 10 open parenthesis t plus 1 close parenthesis cubed plus 10 equals minus 140")[$− 10 ( t + 1 )^(3) + 10 = − 140$] + #math.equation(block: false, alt: "minus 10 open parenthesis t plus 1 close parenthesis cubed plus 10 greater than minus 50")[$− 10 ( t + 1 )^(3) + 10 > − 50$] + #math.equation(block: false, alt: "minus 20 less than minus 10 open parenthesis t plus 1 close parenthesis cubed plus 10 less than 40")[$− 20 < − 10 ( t + 1 )^(3) + 10 < 40$] + #math.equation(block: false, alt: "t approximately equals minus 3.1")[$t ≈ − 3.1$] + #math.equation(block: false, alt: "t approximately equals 1.5")[$t ≈ 1.5$] + #math.equation(block: false, alt: "t less than 0.8")[$t < 0.8$] + #math.equation(block: false, alt: "minus 2.4 less than t less than 0.4")[$− 2.4 < t < 0.4$] The figure shows a graph of #math.equation(block: false, alt: "H open parenthesis z close parenthesis equals 4 the cube root of z minus 4 plus 6")[$H ( z ) = 4 root(3, z − 4) + 6$]. Solve the following: #figure(figph[cube root transformed], alt: "cube root transformed", caption: none) + #math.equation(block: false, alt: "4 the cube root of z minus 4 plus 6 equals 2")[$4 root(3, z − 4) + 6 = 2$] + #math.equation(block: false, alt: "4 the cube root of z minus 4 plus 6 equals 12")[$4 root(3, z − 4) + 6 = 12$] + #math.equation(block: false, alt: "4 the cube root of z minus 4 plus 6 greater than 14")[$4 root(3, z − 4) + 6 > 14$] + #math.equation(block: false, alt: "4 the cube root of z minus 4 plus 6 less than 6")[$4 root(3, z − 4) + 6 < 6$] For Problems 31–34, graph the function with the #strong[ZInteger] setting. Use the graph to solve each equation or inequality. Check your solutions algebraically. Graph #math.equation(block: false, alt: "F open parenthesis x close parenthesis equals 4 the square root of x minus 25")[$" " " " F ( x ) = 4 sqrt(x − 25)$]. + Solve #math.equation(block: false, alt: "4 the square root of x minus 25 equals 16")[$" " " " 4 sqrt(x − 25) = 16$] + Solve #math.equation(block: false, alt: "8 less than 4 the square root of x minus 25 less than or equal to 24")[$" " " " 8 < 4 sqrt(x − 25) ≤ 24$] + #math.equation(block: false, alt: "x equals 41")[$x = 41$] + #math.equation(block: false, alt: "29 less than x less than 61")[$29 < x < 61$] Graph #math.equation(block: false, alt: "G open parenthesis x close parenthesis equals 15 minus 0.01 open parenthesis x minus 2 close parenthesis cubed")[$" " " " G ( x ) = 15 − 0.01 ( x − 2 )^(3)$]. + Solve #math.equation(block: false, alt: "15 minus 0.01 open parenthesis x minus 2 close parenthesis cubed equals minus 18.75")[$" " " " 15 − 0.01 ( x − 2 )^(3) = − 18.75$] + Solve #math.equation(block: false, alt: "15 minus 0.01 open parenthesis x minus 2 close parenthesis cubed less than or equal to 25")[$" " " " 15 − 0.01 ( x − 2 )^(3) ≤ 25$] Graph #math.equation(block: false, alt: "H open parenthesis x close parenthesis equals 24 minus 0.25 open parenthesis x minus 6 close parenthesis squared")[$" " " " H ( x ) = 24 − 0.25 ( x − 6 )^(2)$]. + Solve #math.equation(block: false, alt: "24 minus 0.25 open parenthesis x minus 6 close parenthesis squared equals minus 6.25")[$" " " " 24 − 0.25 ( x − 6 )^(2) = − 6.25$] + Solve #math.equation(block: false, alt: "24 minus 0.25 open parenthesis x minus 6 close parenthesis squared greater than 11.75")[$" " " " 24 − 0.25 ( x − 6 )^(2) > 11.75$] + #math.equation(block: false, alt: "x equals minus 5")[$x = − 5$] or #math.equation(block: false, alt: "x equals 17")[$x = 17$] + #math.equation(block: false, alt: "minus 1 less than x less than 13")[$− 1 < x < 13$] Graph #math.equation(block: false, alt: "R open parenthesis x close parenthesis equals 0.1 open parenthesis x plus 12 close parenthesis squared minus 18")[$" " " " R ( x ) = 0.1 ( x + 12 )^(2) − 18$]. + Solve #math.equation(block: false, alt: "0.1 open parenthesis x plus 12 close parenthesis squared minus 18 equals 14.4")[$" " " " 0.1 ( x + 12 )^(2) − 18 = 14.4$] + Solve #math.equation(block: false, alt: "0.1 open parenthesis x plus 12 close parenthesis squared minus 18 less than 4.5")[$" " " " 0.1 ( x + 12 )^(2) − 18 < 4.5$] For Problems 35–40, + Graph the equation by completing the table and plotting points. + Does the equation define #math.equation(block: false, alt: "y")[$y$] as a function of #math.equation(block: false, alt: "x")[$x$]? Why or why not? #math.equation(block: true, alt: "x equals y squared")[$x = y^(2)$] #figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], )) + #figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction 1 over 4")[$frac(1, 4)$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "4")[$4$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], )) #figure(figph[parabola], alt: "parabola", caption: none) + no #math.equation(block: true, alt: "x equals y cubed")[$x = y^(3)$] #figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], )) #math.equation(block: true, alt: "x equals vertical bar y vertical bar")[$x = | y |$] #figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], )) + #figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], )) #figure(figph[absolute value rotated 90 degrees], alt: "absolute value rotated 90 degrees", caption: none) + no #math.equation(block: true, alt: "x equals vertical bar y vertical bar")[$x = | y |$] #figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], )) #math.equation(block: true, alt: "x equals the fraction 1 over y")[$x = display(frac(1, y))$] #figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], )) + #figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [undefined], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], )) #figure(figph[absolute value rotated 90 degrees], alt: "absolute value rotated 90 degrees", caption: none) + yes #math.equation(block: true, alt: "x equals the fraction 1 over y squared")[$x = display(frac(1, y^(2)))$] #figure(table( columns: 8, align: left, inset: 6pt, table.header([#math.equation(block: false, alt: "x")[$x$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$]), [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "minus the fraction 1 over 2")[$− frac(1, 2)$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "the fraction 1 over 2")[$frac(1, 2)$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], )) For Problems 41–52, graph the following piecewise defined functions. Indicate whether the endpoints of each piece are included on the graph. #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals open brace minus 2, if x less than or equal to 1; x minus 3, if x greater than 1")[$f ( x ) = \{ − 2 & "if " x ≤ 1 \ x − 3 & "if " x > 1$] #figure(figph[piecewise function], alt: "piecewise function", caption: none) #math.equation(block: true, alt: "h open parenthesis x close parenthesis equals open brace minus x plus 2, if x less than or equal to minus 1; 3, if x greater than minus 1")[$h ( x ) = \{ − x + 2 & "if " x ≤ − 1 \ 3 & "if " x > − 1$] #math.equation(block: true, alt: "G open parenthesis t close parenthesis equals open brace 3 t plus 9, if t less than minus 2; minus 3 minus the fraction 1 over 2 t, if t greater than or equal to minus 2")[$G ( t ) = \{ 3 t + 9 & "if " t < − 2 \ − 3 − display(frac(1, 2)) t & "if " t ≥ − 2$] #figure(figph[piecewise function], alt: "piecewise function", caption: none) #math.equation(block: true, alt: "F open parenthesis s close parenthesis equals open brace the fraction 1 over 3 s plus 3, if s less than 3; 2 s minus 3, if s greater than or equal to 3")[$F ( s ) = \{ display(frac(1, 3)) s + 3 & "if " s < 3 \ 2 s − 3 & "if " s ≥ 3$] #math.equation(block: true, alt: "H open parenthesis t close parenthesis equals open brace t squared, if t less than or equal to 1; the fraction 1 over 2 t plus the fraction 1 over 2, if t greater than 1")[$H ( t ) = \{ t^(2) & "if " t ≤ 1 \ display(frac(1, 2)) t + display(frac(1, 2)) & "if " t > 1$] #figure(figph[piecewise function], alt: "piecewise function", caption: none) #math.equation(block: true, alt: "g open parenthesis t close parenthesis equals open brace the fraction 3 over 2 t plus 7, if t less than or equal to minus 2; t squared, if t greater than minus 2")[$g ( t ) = \{ display(frac(3, 2)) t + 7 & "if " t ≤ − 2 \ t^(2) & "if " t > − 2$] #math.equation(block: true, alt: "k open parenthesis x close parenthesis equals open brace vertical bar x vertical bar, if x less than or equal to 2; the square root of x, if x greater than 2")[$k ( x ) = \{ | x | & "if " x ≤ 2 \ sqrt(x) & "if " x > 2$] #figure(figph[piecewise function], alt: "piecewise function", caption: none) #math.equation(block: true, alt: "S open parenthesis x close parenthesis equals open brace the fraction 1 over x, if x less than 1; vertical bar x vertical bar, if x greater than or equal to 1")[$S ( x ) = \{ display(frac(1, x)) & "if " x < 1 \ | x | & "if " x ≥ 1$] #math.equation(block: true, alt: "D open parenthesis x close parenthesis equals open brace vertical bar x vertical bar, if x less than minus 1; x cubed, if x greater than or equal to minus 1")[$D ( x ) = \{ | x | & "if " x < − 1 \ x^(3) & "if " x ≥ − 1$] #figure(figph[piecewise function], alt: "piecewise function", caption: none) #math.equation(block: true, alt: "m open parenthesis x close parenthesis equals open brace x squared, if x less than or equal to the fraction 1 over 2; vertical bar x vertical bar, if x greater than the fraction 1 over 2")[$m ( x ) = \{ x^(2) & "if " x ≤ display(frac(1, 2)) \ | x | & "if " x > display(frac(1, 2))$] #math.equation(block: true, alt: "P open parenthesis t close parenthesis equals open brace t cubed, if t less than or equal to 1; the fraction 1 over t squared, if t greater than 1")[$P ( t ) = \{ t^(3) & "if " t ≤ 1 \ display(frac(1, t^(2))) & "if " t > 1$] #figure(figph[piecewise function], alt: "piecewise function", caption: none) #math.equation(block: true, alt: "Q open parenthesis t close parenthesis equals open brace t squared, if t less than or equal to minus 1; the cube root of t, if t greater than minus 1")[$Q ( t ) = \{ t^(2) & "if " t ≤ − 1 \ root(3, t) & "if " t > − 1$] For Problems 53–58, write a piecewise definition for the function and sketch its graph. #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals vertical bar 2 x minus 8 vertical bar")[$f ( x ) = | 2 x − 8 |$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals open brace 8 minus 2 x, x less than 4; 2 x minus 8, x greater than or equal to 4")[$f ( x ) = \{ 8 − 2 x & x < 4 \ 2 x − 8 & x ≥ 4$] #figure(figph[piecewise function], alt: "piecewise function", caption: none) #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals vertical bar 3 x plus 6 vertical bar")[$g ( x ) = | 3 x + 6 |$] #math.equation(block: true, alt: "g open parenthesis t close parenthesis equals vertical bar 1 plus the fraction t over 3 vertical bar")[$g ( t ) = | 1 + display(frac(t, 3)) |$] #math.equation(block: true, alt: "g open parenthesis t close parenthesis equals open brace minus 1 minus the fraction t over 3, t less than minus 3; 1 plus the fraction t over 3, t greater than or equal to minus 3")[$g ( t ) = \{ − 1 − display(frac(t, 3)) & t < − 3 \ 1 + display(frac(t, 3)) & t ≥ − 3$] #figure(figph[piecewise function], alt: "piecewise function", caption: none) #math.equation(block: true, alt: "f open parenthesis t close parenthesis equals vertical bar the fraction 1 over 2 t minus 3 vertical bar")[$f ( t ) = | display(frac(1, 2)) t − 3 |$] #math.equation(block: true, alt: "F open parenthesis x close parenthesis equals vertical bar x cubed vertical bar")[$F ( x ) = | x^(3) |$] #math.equation(block: true, alt: "F open parenthesis x close parenthesis equals open brace minus x cubed, x less than 0; x cubed, x greater than or equal to 0")[$F ( x ) = \{ − x^(3) & x < 0 \ x^(3) & x ≥ 0$] #figure(figph[piecewise function], alt: "piecewise function", caption: none) #math.equation(block: true, alt: "G open parenthesis x close parenthesis equals vertical bar the fraction 1 over x vertical bar")[$G ( x ) = | display(frac(1, x)) |$] In Problems 59–64, decide whether each statement is true for all values of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]. If the statement is true, give an algebraic justification. If it is false, find values of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] to disprove it. + #math.equation(block: false, alt: "f open parenthesis a plus b close parenthesis equals f open parenthesis a close parenthesis plus f open parenthesis b close parenthesis")[$f ( a + b ) = f ( a ) + f ( b )$] + #math.equation(block: false, alt: "f open parenthesis a b close parenthesis equals f open parenthesis a close parenthesis f open parenthesis b close parenthesis")[$f ( a b ) = f ( a ) f ( b )$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x squared")[$f ( x ) = x^(2)$] + Not always true: #math.equation(block: false, alt: "f open parenthesis 1 plus 2 close parenthesis not equal to f open parenthesis 1 close parenthesis plus f open parenthesis 2 close parenthesis")[$f ( 1 + 2 ) ≠ f ( 1 ) + f ( 2 )$] because #math.equation(block: false, alt: "9 not equal to 5")[$9 ≠ 5$]. + True: #math.equation(block: false, alt: "open parenthesis a b close parenthesis squared equals a squared b squared")[$( a b )^(2) = a^(2) b^(2)$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x cubed")[$f ( x ) = x^(3)$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction 1 over x")[$f ( x ) = display(frac(1, x))$] + Not always true: #math.equation(block: false, alt: "f open parenthesis 1 plus 2 close parenthesis not equal to f open parenthesis 1 close parenthesis plus f open parenthesis 2 close parenthesis")[$f ( 1 + 2 ) ≠ f ( 1 ) + f ( 2 )$] because #math.equation(block: false, alt: "the fraction 1 over 3 not equal to the fraction 3 over 2")[$frac(1, 3) ≠ frac(3, 2)$]. + True: #math.equation(block: false, alt: "the fraction 1 over a b equals the fraction 1 over a times the fraction 1 over b")[$display(frac(1, a b)) = display(frac(1, a)) ⋅ display(frac(1, b))$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of x")[$f ( x ) = sqrt(x)$] #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals m x plus b")[$f ( x ) = m x + b$] + Not always true (unless #math.equation(block: false, alt: "b equals 0")[$b = 0$]): #math.equation(block: false, alt: "f open parenthesis 1 plus 2 close parenthesis not equal to f open parenthesis 1 close parenthesis plus f open parenthesis 2 close parenthesis")[$f ( 1 + 2 ) ≠ f ( 1 ) + f ( 2 )$] because #math.equation(block: false, alt: "3 m plus b not equal to 3 m plus 2 b")[$3 m + b ≠ 3 m + 2 b$]. + Not always true: #math.equation(block: false, alt: "f open parenthesis 1 times 2 close parenthesis not equal to f open parenthesis 1 close parenthesis times f open parenthesis 2 close parenthesis")[$f ( 1 ⋅ 2 ) ≠ f ( 1 ) ⋅ f ( 2 )$] because #math.equation(block: false, alt: "2 m plus b not equal to 2 m squared plus 3 m b plus b squared")[$2 m + b ≠ 2 m^(2) + 3 m b + b^(2)$]. #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals k x")[$f ( x ) = k x$] Verify that #math.equation(block: false, alt: "vertical bar a minus b vertical bar")[$| a − b |$] gives the distance between #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] on a number line. + #math.equation(block: false, alt: "a equals 3 , b equals 8")[$a = 3 , " " b = 8$] + #math.equation(block: false, alt: "a equals minus 2 , b equals minus 6")[$a = − 2 , " " b = − 6$] + #math.equation(block: false, alt: "a equals 4 , b equals minus 3")[$a = 4 , " " b = − 3$] + #math.equation(block: false, alt: "a equals minus 2 , b equals 5")[$a = − 2 , " " b = 5$] + #figure(figph[number line], alt: "number line", caption: none) + #figure(figph[number line], alt: "number line", caption: none) + #figure(figph[number line], alt: "number line", caption: none) + #figure(figph[number line], alt: "number line", caption: none) Which of the following statements is true for all values of #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$]? + #math.equation(block: false, alt: "vertical bar a minus b vertical bar equals vertical bar a vertical bar plus vertical bar b vertical bar")[$| a − b | = | a | + | b |$] + #math.equation(block: false, alt: "vertical bar a minus b vertical bar less than or equal to vertical bar a vertical bar plus vertical bar b vertical bar")[$| a − b | ≤ | a | + | b |$] + #math.equation(block: false, alt: "vertical bar a minus b vertical bar greater than or equal to vertical bar a vertical bar plus vertical bar b vertical bar")[$| a − b | ≥ | a | + | b |$] Explain how the distributive law, #math.equation(block: false, alt: "a open parenthesis b plus c close parenthesis equals a b plus a c")[$a ( b + c ) = a b + a c$], is different from the equation #math.equation(block: false, alt: "f open parenthesis a plus b close parenthesis equals f open parenthesis a close parenthesis plus f open parenthesis b close parenthesis")[$f ( a + b ) = f ( a ) + f ( b )$] . The distributive law shows a relationship between multiplication and addition that always holds. The equation #math.equation(block: false, alt: "f open parenthesis a plus b close parenthesis equals f open parenthesis a close parenthesis plus f open parenthesis b close parenthesis")[$f ( a + b ) = f ( a ) + f ( b )$] is not about multiplication and may or may not be true. For each function, decide whether #math.equation(block: false, alt: "f open parenthesis k x close parenthesis equals k f open parenthesis x close parenthesis")[$f ( k x ) = k f ( x )$] for all #math.equation(block: false, alt: "x not equal to 0")[$x ≠ 0$], where #math.equation(block: false, alt: "k not equal to 0")[$k ≠ 0$] is a constant. + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared")[$f ( x ) = x^(2)$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 1 over x")[$f ( x ) = display(frac(1, x))$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x")[$f ( x ) = sqrt(x)$] + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals vertical bar x vertical bar")[$f ( x ) = | x |$]