#set document(title: "2.6 Domain and Range", 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"))


    
== 2.6#h(0.6em)Domain and Range


    
      
=== Definitions of Domain and Range


      In Example of Graphs of Functions, we graphed the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x plus 4")[$f ( x ) = sqrt(x + 4)$] and observed that #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] is undefined for #math.equation(block: false, alt: "x")[$x$]-values less than #math.equation(block: false, alt: "minus 4")[$− 4$]. For this function, we must choose #math.equation(block: false, alt: "x")[$x$]-values in the interval #math.equation(block: false, alt: "open bracket minus 4 , ∞ close parenthesis")[$[ − 4 , upright(∞) )$].


      #figure(figph[square root graph], alt: "square root graph", caption: none)


      All the points on the graph have #math.equation(block: false, alt: "x")[$x$]-coordinates greater than or equal to #math.equation(block: false, alt: "minus 4")[$− 4$], as shown at left. The set of all permissible values of the input variable is called the #strong[domain] of the function #math.equation(block: false, alt: "f")[$f$].


      We also see that there are no points with negative #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]-values on the graph of #math.equation(block: false, alt: "f")[$f$]: All the points have #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]-values greater than or equal to zero. The set of all outputs or function values corresponding to the domain is called the #strong[range] of the function. Thus, the domain of the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x plus 4")[$f ( x ) = sqrt(x + 4)$] is the interval #math.equation(block: false, alt: "open bracket minus 4 , ∞ close parenthesis")[$[ − 4 , upright(∞) )$], and its range is the interval #math.equation(block: false, alt: "open bracket 0 , ∞ close parenthesis")[$[ 0 , upright(∞) )$]. In general, we make the following definitions.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Domain and Range]


        The #strong[domain] of a function is the set of permissible values for the input variable. The #strong[range] is the set of function values (that is, values of the output variable) that correspond to the domain values.
]


      Using the notions of domain and range, we restate the definition of a function as follows.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Definition of Function]


        A relationship between two variables is a #strong[function] if each element of the domain is paired with exactly one element of the range.
]


      
        In a function, how many domain elements can be paired with one range element?


        \_\_\_\_\_


        #solutionbox[
It is possible that infinitely many domain elements can be paired with one range element.
]


      
      
        In a function, how many domain elements can be paired with one range element?


        + One
+ Two
+ None
+ Infinitely many


      
    
    
      
=== Finding Domain and Range from a Graph


      We can identify the domain and range of a function from its graph. The domain is the set of #math.equation(block: false, alt: "x")[$x$]-values of all points on the graph, and the range is the set of #math.equation(block: false, alt: "y")[$y$]-values.


      #examplebox("Example 1")[][
+ Determine the domain and range of the function #math.equation(block: false, alt: "h")[$h$] graphed at right.
+ Give the function values shown by the indicated points on the graph.


          #figure(figph[graph of function], alt: "graph of function", caption: none)


        
        #solutionbox[
+ All the points on the graph have #math.equation(block: false, alt: "v")[$v$]-coordinates between #math.equation(block: false, alt: "1")[$1$] and #math.equation(block: false, alt: "10")[$10$], inclusive, so the domain of the function #math.equation(block: false, alt: "h")[$h$] is the interval #math.equation(block: false, alt: "open bracket 1 , 10 close bracket")[$[ 1 , 10 ]$]. The #math.equation(block: false, alt: "h open parenthesis v close parenthesis")[$h ( v )$]-coordinates have values between #math.equation(block: false, alt: "minus 2")[$− 2$] and #math.equation(block: false, alt: "7")[$7$], inclusive, so the range of the function is the interval #math.equation(block: false, alt: "open bracket minus 2 , 7 close bracket")[$[ − 2 , 7 ]$].
+ Recall that the points on the graph of a function have coordinates #math.equation(block: false, alt: "open parenthesis v , h open parenthesis v close parenthesis close parenthesis")[$( v , h ( v ) )$]. In other words, the coordinates of each point are made up of a domain value and its corresponding range value. Read the coordinates of the indicated points to obtain the function values #math.equation(block: false, alt: "h open parenthesis 1 close parenthesis equals 3 , h open parenthesis 3 close parenthesis equals minus 2 , h open parenthesis 6 close parenthesis equals minus 1 , h open parenthesis 7 close parenthesis equals 0 , and h open parenthesis 10 close parenthesis equals 7")[$h ( 1 ) = 3 , " " h ( 3 ) = − 2 , " " h ( 6 ) = − 1 , " " h ( 7 ) = 0 , " " "and" " " h ( 10 ) = 7$].
]
]


      #figure(figph[graph of function with enclosing rectangle], alt: "graph of function with enclosing rectangle", caption: none)


      The figure at left shows the graph of the function #math.equation(block: false, alt: "h")[$h$] in Example with the domain values marked on the horizontal axis and the range values marked on the vertical axis.
Imagine a rectangle whose length and width are determined by those segments, as shown in the figure.  All the points #math.equation(block: false, alt: "open parenthesis v , h open parenthesis v close parenthesis close parenthesis")[$( v , h ( v ) )$] on the graph of the function lie within this rectangle.


      The rectangle described above is a convenient window in the plane for viewing the function. Of course, if the domain or range of the function is an infinite interval, we can never include the whole graph within a viewing rectangle and must be satisfied with studying only the important parts of the graph.


      
        #figure(figph[graph of function], alt: "graph of function", caption: none)


        + Draw the smallest viewing window possible around the graph shown above.
+ Find the domain and range of the function. #linebreak() Domain: \_\_\_\_\_  #linebreak() Range: \_\_\_\_\_


        #solutionbox[
A graph is shown below.


          domain: #math.equation(block: false, alt: "open bracket minus 4 , 2 close bracket")[$[ − 4 , 2 ]$]; range: #math.equation(block: false, alt: "open bracket minus 6 , 10.1 close bracket")[$[ − 6 , 10.1 ]$]
]


        #figure(figph[curving touching sides of window], alt: "curving touching sides of window", caption: none)


      
      
        #figure(figph[graph of function], alt: "graph of function", caption: none)


        + Draw the smallest viewing window possible around the graph shown above.
+ Find the domain and range of the function.


        #solutionbox[
+ #figure(figph[curving touching sides of window], alt: "curving touching sides of window", caption: none)
+ domain: #math.equation(block: false, alt: "open bracket minus 4 , 2 close bracket")[$[ − 4 , 2 ]$]; #linebreak() range: #math.equation(block: false, alt: "open bracket minus 6 , 10.1 close bracket")[$[ − 6 , 10.1 ]$]
]


      
      Sometimes the domain is given as part of the definition of a function.


      #examplebox("Example 2")[][
Graph the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals x squared minus 6")[$" " f ( x ) = x^(2) − 6 " "$] on the domain #math.equation(block: false, alt: "0 less than or equal to x less than or equal to 4")[$" " 0 ≤ x ≤ 4$] and give its range.


        
        #solutionbox[
The graph is part of a parabola that opens upward. We obtain several points on the graph by evaluating the function at convenient #math.equation(block: false, alt: "x")[$x$]-values in the domain.


          #figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$" " x " "$]], [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [], []),
  [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 6")[$− 6$]], [], [#math.equation(block: false, alt: "since f open parenthesis 0 close parenthesis equals 0 squared minus 6 equals minus 6")[$"since " f ( 0 ) = 0^(2) − 6 = − 6$]],
  [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "minus 5")[$− 5$]], [], [#math.equation(block: false, alt: "since f open parenthesis 1 close parenthesis equals 1 squared minus 6 equals minus 5")[$"since " f ( 1 ) = 1^(2) − 6 = − 5$]],
  [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [], [#math.equation(block: false, alt: "since f open parenthesis 2 close parenthesis equals 2 squared minus 6 equals minus 2")[$"since " f ( 2 ) = 2^(2) − 6 = − 2$]],
  [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "3")[$3$]], [], [#math.equation(block: false, alt: "since f open parenthesis 3 close parenthesis equals 3 squared minus 6 equals 3")[$"since " f ( 3 ) = 3^(2) − 6 = 3$]],
  [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "10")[$10$]], [], [#math.equation(block: false, alt: "since f open parenthesis 4 close parenthesis equals 4 squared minus 6 equals 10")[$"since " f ( 4 ) = 4^(2) − 6 = 10$]],
))


          The range of the function is the set of all #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]-values that appear on the graph. We can see  that the lowest point on the graph is #math.equation(block: false, alt: "open parenthesis 0 , minus 6 close parenthesis")[$( 0 , − 6 )$], so the smallest #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]-value is #math.equation(block: false, alt: "minus 6")[$− 6$]. The highest point on the graph is #math.equation(block: false, alt: "open parenthesis 4 , 10 close parenthesis")[$( 4 , 10 )$], so the largest #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]-value is #math.equation(block: false, alt: "10")[$10$]. Thus, the range of the function #math.equation(block: false, alt: "f")[$f$] is the interval #math.equation(block: false, alt: "open bracket minus 6 , 10 close bracket")[$[ − 6 , 10 ]$].


          #figure(figph[parabola with x from 0 to 4], alt: "parabola with x from 0 to 4", caption: none)
]
]


      
        Graph the function #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x cubed minus 4")[$g ( x ) = x^(3) − 4$] on the domain #math.equation(block: false, alt: "open bracket minus 2 , 3 close bracket")[$[ − 2 , 3 ]$] and give its range.


        Range: \_\_\_\_\_


        #solutionbox[
A graph is below.


          range: #math.equation(block: false, alt: "open bracket minus 12 , 23 close bracket")[$[ − 12 , 23 ]$]
]


        A graph of #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x cubed minus 4")[$g ( x ) = x^(3) − 4$] on the domain #math.equation(block: false, alt: "open bracket minus 2 , 3 close bracket")[$[ − 2 , 3 ]$].


        #figure(figph[cubic on finite domain], alt: "cubic on finite domain", caption: none)


      
      
        Graph the function #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals x cubed minus 4")[$" " g ( x ) = x^(3) − 4 " "$] on the domain #math.equation(block: false, alt: "open bracket minus 2 , 3 close bracket")[$[ − 2 , 3 ]$] and give its range.


        #solutionbox[
#figure(figph[cubic on finite domain], alt: "cubic on finite domain", caption: none)


          range: #math.equation(block: false, alt: "open bracket minus 12 , 23 close bracket")[$[ − 12 , 23 ]$]
]


      
      
        Which function changes concavity on its domain?


        \_\_\_\_\_


        #solutionbox[
The function #math.equation(block: false, alt: "y equals x cubed")[$y = x^(3)$] changes concavity on its domain.
]


      
      
        Which function changes concavity on its domain?


        + #math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$]
+ #math.equation(block: false, alt: "y equals x cubed")[$y = x^(3)$]
+ #math.equation(block: false, alt: "y equals the square root of x")[$y = sqrt(x)$]
+ #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$]


      
      Not all functions have domains and ranges that are intervals.


      #examplebox("Example 3")[][
+ The table gives the postage for sending printed material by first-class mail in 2016. Graph the postage function #math.equation(block: false, alt: "p equals g open parenthesis w close parenthesis")[$p = g ( w )$].#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Weight in ounces #math.equation(block: false, alt: "open parenthesis w close parenthesis")[$( w )$]], [Postage #math.equation(block: false, alt: "open parenthesis p close parenthesis")[$( p )$]]),
  [#math.equation(block: false, alt: "0 less than w less than or equal to 1")[$0 < w ≤ 1$]], [\$0.47],
  [#math.equation(block: false, alt: "1 less than w less than or equal to 2")[$1 < w ≤ 2$]], [\$0.68],
  [#math.equation(block: false, alt: "2 less than w less than or equal to 3")[$2 < w ≤ 3$]], [\$0.89],
  [#math.equation(block: false, alt: "3 less than w less than or equal to 4")[$3 < w ≤ 4$]], [\$1.10],
  [#math.equation(block: false, alt: "4 less than w less than or equal to 5")[$4 < w ≤ 5$]], [\$1.31],
  [#math.equation(block: false, alt: "5 less than w less than or equal to 6")[$5 < w ≤ 6$]], [\$1.52],
  [#math.equation(block: false, alt: "6 less than w less than or equal to 7")[$6 < w ≤ 7$]], [\$1.73],
))
+ Determine the domain and range of the function.


        
        #solutionbox[
+ From the table, we see that articles of any weight up to #math.equation(block: false, alt: "1")[$1$] ounce require \$0.47 postage. This means that for all #math.equation(block: false, alt: "w")[$w$]-values greater than #math.equation(block: false, alt: "0")[$0$] but less than or equal to #math.equation(block: false, alt: "1")[$1$], the #math.equation(block: false, alt: "p")[$p$]-value is #math.equation(block: false, alt: "0.47")[$0.47$]. Thus, the graph of #math.equation(block: false, alt: "p equals g open parenthesis w close parenthesis")[$p = g ( w )$] between #math.equation(block: false, alt: "w equals 0")[$w = 0$] and #math.equation(block: false, alt: "w equals 1")[$w = 1$] looks like a small piece of the horizontal line #math.equation(block: false, alt: "p equals 0.47")[$p = 0.47$].  #linebreak() Similarly, for all #math.equation(block: false, alt: "w")[$w$]-values greater than #math.equation(block: false, alt: "1")[$1$] but less than or equal to #math.equation(block: false, alt: "2")[$2$], the #math.equation(block: false, alt: "p")[$p$]-value is #math.equation(block: false, alt: "0.68")[$0.68$], so the graph on this interval looks like a small piece of the line #math.equation(block: false, alt: "p equals 0.68")[$p = 0.68$]. Continue in this way to obtain the graph shown below.#figure(figph[postage step function], alt: "postage step function", caption: none)

 #linebreak() The open circles at the left endpoint of each horizontal segment indicate that that point is not included in the graph; the closed circles are points on the graph. For instance, if #math.equation(block: false, alt: "w equals 3")[$w = 3$], the postage, #math.equation(block: false, alt: "p")[$p$], is \$0.89, not \$1.10. Consequently, the point #math.equation(block: false, alt: "open parenthesis 3 , 0.89 close parenthesis")[$( 3 , 0.89 )$] is part of the graph of #math.equation(block: false, alt: "g")[$g$], but the point #math.equation(block: false, alt: "open parenthesis 3 , 1.10 close parenthesis")[$( 3 , 1.10 )$] is not.
+ Postage rates are given for all weights greater than #math.equation(block: false, alt: "0")[$0$] ounces up to and including #math.equation(block: false, alt: "7")[$7$] ounces, so the domain of the function is the half-open interval #math.equation(block: false, alt: "open parenthesis 0 , 7 close bracket")[$( 0 , 7 ]$]. (The domain is an interval because there is a point on the graph for every #math.equation(block: false, alt: "w")[$w$]-value from #math.equation(block: false, alt: "0")[$0$] to #math.equation(block: false, alt: "7")[$7$].) #linebreak()  The range of the function is not an interval, however, because the possible values for #math.equation(block: false, alt: "p")[$p$] do not include all the real numbers between #math.equation(block: false, alt: "0.3")[$0.3$] and #math.equation(block: false, alt: "1.75")[$1.75$]. The range is the set of discrete values #math.equation(block: false, alt: "0.47")[$0.47$], #math.equation(block: false, alt: "0.68")[$0.68$], #math.equation(block: false, alt: "0.89")[$0.89$], #math.equation(block: false, alt: "1.10")[$1.10$], #math.equation(block: false, alt: "1.31")[$1.31$], #math.equation(block: false, alt: "1.52")[$1.52$], and #math.equation(block: false, alt: "1.73")[$1.73$].
]
]


      
        Which function includes all real numbers in its range?


        \_\_\_\_\_


        #solutionbox[
The function #math.equation(block: false, alt: "y equals x cubed")[$y = x^(3)$] includes all real numbers in its range.
]


      
      
        Which function includes all real numbers in its range?


        + #math.equation(block: false, alt: "y equals x squared")[$y = x^(2)$]
+ #math.equation(block: false, alt: "y equals x cubed")[$y = x^(3)$]
+ #math.equation(block: false, alt: "y equals the square root of x")[$y = sqrt(x)$]
+ #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$]


      
      
        In Practice 4 of Functions as Mathematical Models, you wrote a formula for residential water bills, #math.equation(block: false, alt: "B open parenthesis w close parenthesis")[$B ( w )$], in Arid, New Mexico:


        #math.equation(block: true, alt: "B open parenthesis w close parenthesis equals open brace 30 plus 2 w ,, 0 less than or equal to w less than or equal to 50; 50 plus 3 w ,, w greater than 50")[$B ( w ) = \{ 30 + 2 w ", " & 0 ≤ w ≤ 50 \ 50 + 3 w ", " & w > 50$]
        If the utilities commission imposes a cap on monthly water consumption at #math.equation(block: false, alt: "120")[$120$] HCF, find the domain and range of the function #math.equation(block: false, alt: "B open parenthesis w close parenthesis")[$B ( w )$].


        Domain: \_\_\_\_\_


        Range: \_\_\_\_\_


        #solutionbox[
domain: #math.equation(block: false, alt: "open bracket 0 , 120 close bracket")[$[ 0 , 120 ]$]; range: #math.equation(block: false, alt: "open bracket 30 , 130 close bracket union open parenthesis 200 , 410 close bracket")[$[ 30 , 130 ] ∪ ( 200 , 410 ]$]
]


      
      
        In Practice 4 of Functions as Mathematical Models, you wrote a formula for residential water bills, #math.equation(block: false, alt: "B open parenthesis w close parenthesis")[$B ( w )$], in Arid, New Mexico:


        #math.equation(block: true, alt: "B open parenthesis w close parenthesis equals open brace 30 plus 2 w ,, 0 less than or equal to w less than or equal to 50; 50 plus 3 w ,, w greater than 50")[$B ( w ) = \{ 30 + 2 w ", " & 0 ≤ w ≤ 50 \ 50 + 3 w ", " & w > 50$]
        If the utilities commission imposes a cap on monthly water consumption at #math.equation(block: false, alt: "120")[$120$] HCF, find the domain and range of the function #math.equation(block: false, alt: "B open parenthesis w close parenthesis")[$B ( w )$].


        #solutionbox[
domain: #math.equation(block: false, alt: "open bracket 0 , 120 close bracket")[$[ 0 , 120 ]$]; range: #math.equation(block: false, alt: "open bracket 30 , 130 close bracket union open parenthesis 200 , 410 close bracket")[$[ 30 , 130 ] ∪ ( 200 , 410 ]$]
]


      
      
        Do the largest and smallest values in the domain of a function result in the largest and smallest values of the range?  Give an example.


        \_\_\_\_\_


      
      
        Do the largest and smallest values in the domain of a function result in the largest and smallest values of the range? Give an example.


      
    
    
      
=== Finding the Domain from a Formula


      If the domain of a function is not given as part of its definition, we assume that the domain is as large as possible. We include in the domain all #math.equation(block: false, alt: "x")[$x$]-values that make sense when substituted into the function's formula.


      For example, the domain of the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of 9 minus x squared")[$" " f ( x ) = sqrt(9 − x^(2)) " "$]  is the interval #math.equation(block: false, alt: "open bracket minus 3 , 3 close bracket")[$[ − 3 , 3 ]$], because #math.equation(block: false, alt: "x")[$x$]-values less than #math.equation(block: false, alt: "minus 3")[$− 3$] or greater than #math.equation(block: false, alt: "3")[$3$] result in square roots of negative numbers. You may recognize the graph of #math.equation(block: false, alt: "f")[$f$] as the upper half of the circle #math.equation(block: false, alt: "x squared plus y squared equals 9")[$" " x^(2) + y^(2) = 9$], as shown at right.


      #figure(figph[graph of upper half circle], alt: "graph of upper half circle", caption: none)


      
        The domain of the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] is #math.equation(block: false, alt: "open bracket minus 4 , 4 close bracket")[$[ − 4 , 4 ]$]. What is the domain of the function #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 1 over 2 f open parenthesis x close parenthesis")[$g ( x ) = display(frac(1, 2)) f ( x )$]?


        \_\_\_\_\_


        #solutionbox[
The domain of #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$] is #math.equation(block: false, alt: "open bracket minus 4 , 4 close bracket")[$[ − 4 , 4 ]$]
]


      
      
        The domain of the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] is #math.equation(block: false, alt: "open bracket minus 4 , 4 close bracket")[$[ − 4 , 4 ]$]. What is the domain of the function #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 1 over 2 f open parenthesis x close parenthesis")[$" " g ( x ) = display(frac(1, 2)) f ( x )$]?


        + #math.equation(block: false, alt: "open bracket minus 2 , 2 close bracket")[$[ − 2 , 2 ]$]
+ #math.equation(block: false, alt: "open bracket minus 8 , 8 close bracket")[$[ − 8 , 8 ]$]
+ #math.equation(block: false, alt: "open bracket minus 4 , 4 close bracket")[$[ − 4 , 4 ]$]
+ #math.equation(block: false, alt: "open bracket 4 , minus 4 close bracket")[$[ 4 , − 4 ]$]


      
      #examplebox("Example 4")[][
Find the domain of the function #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 1 over x minus 3")[$" " " " g ( x ) = display(frac(1, x − 3))$]


        
        #solutionbox[
We must omit any #math.equation(block: false, alt: "x")[$x$]-values that do not make sense in the function's formula. Because division by zero is undefined, we cannot allow the denominator of #math.equation(block: false, alt: "the fraction 1 over x minus 3")[$display(frac(1, x − 3))$] to be zero. Since #math.equation(block: false, alt: "x minus 3 equals 0")[$x − 3 = 0$] when #math.equation(block: false, alt: "x equals 3")[$x = 3$], we exclude #math.equation(block: false, alt: "x equals 3")[$x = 3$] from the domain of #math.equation(block: false, alt: "g")[$g$]. Thus, the domain of #math.equation(block: false, alt: "g")[$g$] is the set of all real numbers except #math.equation(block: false, alt: "3")[$3$].
]
]


      
        + Find the domain of the function #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the fraction 1 over open parenthesis x minus 4 close parenthesis squared")[$h ( x ) = display(frac(1, ( x − 4 )^(2)))$]. #linebreak() Domain: \_\_\_\_\_
+ Graph the function in the window
                    #math.equation(block: true, alt: "Xmin, equals minus 2, Xmax equals 8; Ymin, equals minus 2, Ymax equals 8")[$"Xmin" & = − 2 &  & "Xmax" = 8 \ "Ymin" & = − 2 &  & "Ymax" = 8$]
                    Use your graph and the function's formula to find its range. #linebreak() Range: \_\_\_\_\_


        Note: This online grading system will not recognize the #math.equation(block: false, alt: "not equal to")[$≠$] symbol, only #math.equation(block: false, alt: "less than")[$<$], #math.equation(block: false, alt: "less than or equal to")[$≤$], #math.equation(block: false, alt: "greater than")[$>$], and #math.equation(block: false, alt: "greater than or equal to")[$≥$]. So instead of #math.equation(block: false, alt: "x not equal to 0")[$x ≠ 0$], you could use either the inequalities "x \< 0 or x \> 0" or the intervals "(-inf,0) U (0,inf)".


        #solutionbox[
+ Domain:  #math.equation(block: false, alt: "x not equal to 4")[$x ≠ 4$] #linebreak() A graph is shown below.
+ Range: #math.equation(block: false, alt: "y greater than 0")[$y > 0$]
]


        A graph for part (a):


        #figure(figph[graphing calculator graph], alt: "graphing calculator graph", caption: none)


      
      
        + Find the domain of the function #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the fraction 1 over open parenthesis x minus 4 close parenthesis squared")[$h ( x ) = display(frac(1, ( x − 4 )^(2)))$].
+ Graph the function in the window
                    #math.equation(block: true, alt: "Xmin, equals minus 2, Xmax equals 8; Ymin, equals minus 2, Ymax equals 8")[$"Xmin" & = − 2 &  & "Xmax" = 8 \ "Ymin" & = − 2 &  & "Ymax" = 8$]
            Use your graph and the function's formula to find its range.


        #solutionbox[
+ Domain:  #math.equation(block: false, alt: "x not equal to 4")[$x ≠ 4$]
+ #figure(figph[graphing calculator graph], alt: "graphing calculator graph", caption: none)

 #linebreak() Range: #math.equation(block: false, alt: "y greater than 0")[$y > 0$]
]


      
      For the functions we have studied so far, there are only two operations we must avoid when finding the domain: division by zero and taking the square root of a negative number.


      Many common functions have as their domain the entire set of real numbers. In particular, a linear function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals b plus m x")[$" " f ( x ) = b + m x " "$] can be evaluated at any real number value of #math.equation(block: false, alt: "x")[$x$], so its domain is the set of all real numbers. This set is represented in interval notation as #math.equation(block: false, alt: "open parenthesis minus ∞ , ∞ close parenthesis")[$( − upright(∞) , upright(∞) )$].


      The range of the linear function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals b plus m x")[$" " f ( x ) = b + m x " "$] (if #math.equation(block: false, alt: "m not equal to 0")[$m ≠ 0$]) is also the set of all real numbers, because the graph continues infinitely at both ends, as shown in figure (a). If #math.equation(block: false, alt: "m equals 0")[$m = 0$], then #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals b")[$" " f ( x ) = b " "$], and the graph of #math.equation(block: false, alt: "f")[$f$] is a horizontal line. In this case, the range consists of a single number, #math.equation(block: false, alt: "b")[$b$].


      #figure(figph[increasing line and horizontal line], alt: "increasing line and horizontal line", caption: none)


      
        Which operations must we examine when finding the domain of a function?


        \_\_\_\_\_


        #solutionbox[
We must examine square roots and division by zero when finding the domain of a function defined by a formula.
]


      
      
        Which operations must we examine when finding the domain of a function?


        + absolute value and square roots
+ square roots and cube roots
+ square roots and division by zero
+ absolute value and division by zero


      
      
        State the domain and range of each function:


        #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 4 , g open parenthesis x close parenthesis equals 4 x , h open parenthesis x close parenthesis equals 4 x squared")[$f ( x ) = 4 , " " g ( x ) = 4 x , " " h ( x ) = 4 x^(2)$]
        \_\_\_\_\_


      
      
        State the domain and range of each function:  #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 4 , g open parenthesis x close parenthesis equals 4 x , h open parenthesis x close parenthesis equals 4 x squared")[$f ( x ) = 4 , " " g ( x ) = 4 x , " " h ( x ) = 4 x^(2)$]


      
    
    
      
=== Restricting the Domain


      In many applications, we may restrict the domain of a function to suit the situation at hand.


      #examplebox("Example 5")[][
The function #math.equation(block: false, alt: "h equals f open parenthesis t close parenthesis equals 1454 minus 16 t squared")[$" " h = f ( t ) = 1454 − 16 t^(2) " "$] gives the height of an algebra book dropped from the top of the Sears Tower as a function of time. Give a suitable domain for this application, and the corresponding range.


        
        #solutionbox[
You can use the window


          #math.equation(block: true, alt: "Xmin, equals minus 10, Xmax equals 10; Ymin, equals minus 100, Ymax equals 1500")[$"Xmin" & = − 10 &  & "Xmax" = 10 \ "Ymin" & = − 100 &  & "Ymax" = 1500$]
          to obtain the graph shown at right.


          #figure(figph[calculator graph of parabola], alt: "calculator graph of parabola", caption: none)


          Because #math.equation(block: false, alt: "t")[$t$] represents the time in seconds after the book was dropped, only positive #math.equation(block: false, alt: "t")[$t$]-values make sense for the problem. The book stops falling when it hits the ground, at #math.equation(block: false, alt: "h equals 0")[$h = 0$]. You can verify that this happens at approximately #math.equation(block: false, alt: "t equals 9.5")[$t = 9.5$] seconds. Thus, only #math.equation(block: false, alt: "t")[$t$]-values between #math.equation(block: false, alt: "0")[$0$] and #math.equation(block: false, alt: "9.5")[$9.5$] are realistic for this application, so we restrict the domain of the function #math.equation(block: false, alt: "f")[$f$] to the interval #math.equation(block: false, alt: "open bracket 0 , 9.5 close bracket")[$[ 0 , 9.5 ]$].


          During that time period, the height, #math.equation(block: false, alt: "h")[$h$], of the book decreases from #math.equation(block: false, alt: "1454")[$1454$] feet to #math.equation(block: false, alt: "0")[$0$] feet. The range of the function on the domain #math.equation(block: false, alt: "open bracket 0 , 9.5 close bracket")[$[ 0 , 9.5 ]$] is #math.equation(block: false, alt: "open bracket 0 , 1454 close bracket")[$[ 0 , 1454 ]$]. The graph is shown at right.


          #figure(figph[half of downward opening parabola], alt: "half of downward opening parabola", caption: none)
]
]


      
        #figure(figph[diagram for making box], alt: "diagram for making box", caption: none)


        (See  Geometry formulas for the formula for the volume of a box.)


        The children in Francine's art class are going to make cardboard boxes. Each child is given a sheet of cardboard that measures 18 inches by 24 inches. To make a box, the child will cut out a square from each corner and turn up the edges, as shown below.


        + Write a formula #math.equation(block: false, alt: "V equals f open parenthesis x close parenthesis")[$V = f ( x )$] for the volume of the box in terms of #math.equation(block: false, alt: "x")[$x$], the side of the cut-out square.  #linebreak() #math.equation(block: false, alt: "V equals")[$V =$]\_\_\_\_\_
+ What is the domain of the function? (What are the largest and smallest possible values of #math.equation(block: false, alt: "x")[$x$]?) #linebreak() Domain: \_\_\_\_\_
+ Graph the function and estimate its range. #linebreak() Range: \_\_\_\_\_


        #solutionbox[
+ #math.equation(block: false, alt: "V equals f open parenthesis x close parenthesis equals x open parenthesis 24 minus 2 x close parenthesis open parenthesis 18 minus 2 x close parenthesis")[$V = f ( x ) = x ( 24 − 2 x ) ( 18 − 2 x )$]
+ #math.equation(block: false, alt: "open parenthesis 0 , 9 close parenthesis")[$( 0 , 9 )$]
+ A graph is below. The range is approximately #math.equation(block: false, alt: "open parenthesis 0 , 655 close parenthesis")[$( 0 , 655 )$]
]


        A graph for part (c):


        #figure(figph[part of cubic], alt: "part of cubic", caption: none)


      
      
        The children in Francine's art class are going to make cardboard boxes. Each child is given a sheet of cardboard that measures 18 inches by 24 inches. To make a box, the child will cut out a square from each corner and turn up the edges, as shown below.


        #figure(figph[diagram for making box], alt: "diagram for making box", caption: none)


        + Write a formula #math.equation(block: false, alt: "V equals f open parenthesis x close parenthesis")[$V = f ( x )$] for the volume of the box in terms of #math.equation(block: false, alt: "x")[$x$], the side of the cut-out square. 
        (See Geometry formulas for the formula for the volume of a box.)
+ What is the domain of the function? (What are the largest and smallest possible values of #math.equation(block: false, alt: "x")[$x$]?)
+ Graph the function and estimate its range.


        #solutionbox[
+ #math.equation(block: false, alt: "V equals f open parenthesis x close parenthesis equals x open parenthesis 24 minus 2 x close parenthesis open parenthesis 18 minus 2 x close parenthesis")[$V = f ( x ) = x ( 24 − 2 x ) ( 18 − 2 x )$]
+ #math.equation(block: false, alt: "open parenthesis 0 , 9 close parenthesis")[$( 0 , 9 )$]
+ #figure(figph[part of cubic], alt: "part of cubic", caption: none)

 #linebreak() The range is approximately #math.equation(block: false, alt: "open parenthesis 0 , 655 close parenthesis")[$( 0 , 655 )$]
]


      
    
    
      
=== Section Summary


      
        
==== Vocabulary


        Look up the definitions of new terms in the Glossary.


        - Domain
- Range
- Restricted domain


      
      
        
==== CONCEPTS


        + The #strong[domain] of a function is the set of permissible values for the input variable.
+ The #strong[range] is the set of function values (that is, values of the output variable) that correspond to the domain values.
+ A relationship between two variables is a #strong[function] if each element of the domain is paired with only one element of the range.
+ We can identify the domain and range of a function from its graph. The domain is the set of #math.equation(block: false, alt: "x")[$x$]-values of all points on the graph, and the range is the set of #math.equation(block: false, alt: "y")[$y$]-values.
+ If the domain of a function is not given as part of its definition, we assume that the domain is as large as possible.
+ In applications, we may restrict the domain and range of a function to suit the situation at hand.


      
      
        
==== STUDY QUESTIONS


        + Explain how to find the domain and range of a function from its graph.
+ What is the domain of the function #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 4")[$f ( x ) = 4$]? What is its range?
+ Which of the eight basic functions are increasing on their entire domain? Which are decreasing on their entire domain?
+ Which of the eight basic functions are concave up on their entire domain? Which are concave down on their entire domain?
+ Which of the eight basic functions can be evaluated at any real number? Which can take on any real number as a function value?
+ Which of the eight basic functions can be graphed in one piece, without lifting the pencil from the paper?


      
      
        
==== SKILLS


        Practice each skill in the Homework problems listed.


        + Find the domain and range of a function from its graph: \#1–16
+ Restrict the domain of a function to suit an application: \#17–24
+ Find the domain of a function from its algebraic formula: \#25–30
+ Find the corresponding domain value for a given range value: \#31–38
+ Find the range of a function on a given domain: \#39–50


      
    
    
      
=== Homework 2.6


      For Problems 1–8, find the domain and range of the function from its graph.


      
        #figure(figph[curve], alt: "curve", caption: none)


        
          Domain: #math.equation(block: false, alt: "open bracket minus 5 , 3 close bracket")[$[ − 5 , 3 ]$]; Range: #math.equation(block: false, alt: "open bracket minus 3 , 7 close bracket")[$[ − 3 , 7 ]$]


        
      
      
        #figure(figph[curve], alt: "curve", caption: none)


      
      
        #figure(figph[curve], alt: "curve", caption: none)


        
          Domain: #math.equation(block: false, alt: "open bracket minus 4 , 5 close bracket")[$[ − 4 , 5 ]$]; Range: #math.equation(block: false, alt: "open bracket minus 1 , 1 close parenthesis union open bracket 3 , 6 close bracket")[$[ − 1 , 1 ) ∪ [ 3 , 6 ]$]


        
      
      
        #figure(figph[curve], alt: "curve", caption: none)


      
      
        #figure(figph[curve], alt: "curve", caption: none)


        
          Domain: #math.equation(block: false, alt: "open bracket minus 2 , 2 close bracket")[$[ − 2 , 2 ]$]; Range: #math.equation(block: false, alt: "open bracket minus 1 , 1 close bracket")[$[ − 1 , 1 ]$]


        
      
      
        #figure(figph[curve], alt: "curve", caption: none)


      
      
        #figure(figph[curve], alt: "curve", caption: none)


        
          Domain: #math.equation(block: false, alt: "open parenthesis minus 5 , 5 close bracket")[$( − 5 , 5 ]$]; Range: #math.equation(block: false, alt: "open brace minus 1 , 0 , 2 , 3 close brace")[$\{ − 1 , 0 , 2 , 3 \}$]


        
      
      
        #figure(figph[curve], alt: "curve", caption: none)


      
      For Problems 9–2, state the domain and range of the basic function.


      
        + #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 squared")[$g ( x ) = x^(2)$]


        
          + Domain: all real numbers; Range: all real numbers
+ Domain: all real numbers; Range: #math.equation(block: false, alt: "open bracket 0 , ∞ close parenthesis")[$[ 0 , upright(∞) )$]


        
      
      
        + #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 x")[$G ( x ) = x$]


      
      
        + #math.equation(block: false, alt: "H open parenthesis x close parenthesis equals the fraction 1 over x squared")[$H ( x ) = display(frac(1, x^(2)))$]
+ #math.equation(block: false, alt: "M open parenthesis x close parenthesis equals the fraction 1 over x")[$M ( x ) = display(frac(1, x))$]


        
          + Domain: all real numbers except zero; Range: #math.equation(block: false, alt: "open parenthesis 0 , ∞ close parenthesis")[$( 0 , upright(∞) )$]
+ Domain: all real numbers except zero; Range: all real numbers except zero


        
      
      
        + #math.equation(block: false, alt: "p open parenthesis x close parenthesis equals the cube root of x")[$p ( x ) = root(3, x)$]
+ #math.equation(block: false, alt: "q open parenthesis x close parenthesis equals the square root of x")[$q ( x ) = sqrt(x)$]


      
      
        The graph shows the elevation of the Los Angeles Marathon course as a function of the distance into the race, #math.equation(block: false, alt: "a equals f open parenthesis t close parenthesis")[$a = f ( t )$]. Estimate the domain and range of the function. (Source: Los Angeles Times)


        #figure(figph[LA marathon elevation], alt: "LA marathon elevation", caption: none)


        
          Domain: #math.equation(block: false, alt: "open bracket 0 , 26.2 close bracket")[$[ 0 , 26.2 ]$]; Range: #math.equation(block: false, alt: "open bracket 90 , 300 close bracket")[$[ 90 , 300 ]$]


        
      
      
        The graph shows the federal debt as a percentage of the gross domestic product, as a function of time, #math.equation(block: false, alt: "D equals f open parenthesis t close parenthesis")[$D = f ( t )$]. Estimate the domain and range of the function. (Source: Office of Management and Budget)


        #figure(figph[US debt], alt: "US debt", caption: none)


      
      
        The graph shows the average air temperature as a function of altitude, #math.equation(block: false, alt: "T equals f open parenthesis h close parenthesis")[$T = f ( h )$]. Estimate the domain and range of the function. (Source: Ahrens, 1998)


        #figure(figph[temperature vs altitude], alt: "temperature vs altitude", caption: none)


        
          Domain: #math.equation(block: false, alt: "open bracket 0 , 600 close bracket")[$[ 0 , 600 ]$]; Range: #math.equation(block: false, alt: "open bracket minus 90 , 700 close bracket")[$[ − 90 , 700 ]$]


        
      
      
        The graph shows the speed of sound in the ocean as a function of depth, #math.equation(block: false, alt: "S equals f open parenthesis d close parenthesis")[$S = f ( d )$]. Estimate the domain and range of the function. (Source: Scientific American)


        #figure(figph[speed of sound vs depth], alt: "speed of sound vs depth", caption: none)


      
      
        Clinton purchases \$#math.equation(block: false, alt: "6000")[$6000$] of photographic equipment to set up his studio. He estimates a salvage value of \$#math.equation(block: false, alt: "500")[$500$] for the equipment in #math.equation(block: false, alt: "10")[$10$] years, and for tax purposes he uses straight-line depreciation.


        + Write a formula for the value of the equipment, #math.equation(block: false, alt: "V open parenthesis t close parenthesis")[$V ( t )$], after #math.equation(block: false, alt: "t")[$t$] years.
+ State the domain and range of the function #math.equation(block: false, alt: "V open parenthesis t close parenthesis")[$V ( t )$].


        
          + #math.equation(block: false, alt: "V open parenthesis t close parenthesis equals 6000 minus 550 t")[$V ( t ) = 6000 − 550 t$]
+ Domain: #math.equation(block: false, alt: "open bracket 0 , 10 close bracket")[$[ 0 , 10 ]$]; Range: #math.equation(block: false, alt: "open bracket 500 , 6000 close bracket")[$[ 500 , 6000 ]$]


        
      
      
        Leslie plans to invest some money in two CD accounts. The first account pays #math.equation(block: false, alt: "3.6 %")[$3.6 upright(%)$] interest per year, and the second account pays #math.equation(block: false, alt: "2.8 %")[$2.8 upright(%)$] interest per year. Leslie would like to earn \$#math.equation(block: false, alt: "500")[$500$] per year on her investment.


        + Write a linear equation in general form that relates #math.equation(block: false, alt: "x")[$x$], the amount Leslie invests at #math.equation(block: false, alt: "3.4 %")[$3.4 upright(%)$], and #math.equation(block: false, alt: "y")[$y$], the amount she invests at #math.equation(block: false, alt: "2.8 %")[$2.8 upright(%)$].
+ Use your equation from part (a) to write #math.equation(block: false, alt: "y")[$y$] as a function of #math.equation(block: false, alt: "x")[$x$], #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$].
+ Find the domain and range of #math.equation(block: false, alt: "f")[$f$].


      
      
        The height of a golfball, in feet, #math.equation(block: false, alt: "t")[$t$] seconds after being hit is given by the function #math.equation(block: false, alt: "h equals f open parenthesis t close parenthesis equals minus 16 open parenthesis t minus 2 close parenthesis squared plus 64")[$h = f ( t ) = − 16 ( t − 2 )^(2) + 64$].


        + Graph the function.
+ State the domain and range of the function and explain what they tell us about the golfball.


        
          + #figure(figph[parabola], alt: "parabola", caption: none)
+ Domain: #math.equation(block: false, alt: "open bracket 0 , 4 close bracket")[$[ 0 , 4 ]$]; Range: #math.equation(block: false, alt: "open bracket 0 , 64 close bracket .")[$[ 0 , 64 ] . " " " "$]The ball reaches a height of 64 feet and hits the ground 4 seconds after being hit.


        
      
      
        Gameworld is marketing a new boardgame called Synaps. If Gameworld charges #math.equation(block: false, alt: "p")[$p$] dollars for the game, their revenue is given by the function #math.equation(block: false, alt: "R equals f open parenthesis p close parenthesis equals minus 50 open parenthesis p minus 10 close parenthesis squared plus 5000")[$R = f ( p ) = − 50 ( p − 10 )^(2) + 5000$].


        + Graph the function.
+ State the domain and range of the function and explain what they tell us about the revenue.


      
      
        In New York City, taxi cabs charge \$2.50 for distances up to #math.equation(block: false, alt: "the fraction 1 over 3")[$display(frac(1, 3))$] mile, plus \$0.40 for each additional #math.equation(block: false, alt: "the fraction 1 over 5")[$display(frac(1, 5))$] mile or portion thereof. (Source: www.visitnyc.com)


        + Sketch a graph of #math.equation(block: false, alt: "F open parenthesis d close parenthesis")[$F ( d )$], which gives taxi fare as a function of distance traveled, on the domain #math.equation(block: false, alt: "0 less than d less than 1")[$0 < d < 1$].
+ State the range of #math.equation(block: false, alt: "F open parenthesis d close parenthesis")[$F ( d )$] on that domain.
+ How much will it cost Renee to travel by taxi from Columbia University to Rockefeller Center, a distance of 5.7 miles?


        
          + #figure(figph[step function], alt: "step function", caption: none)
+ Range: #math.equation(block: false, alt: "open brace 2.50 , 2.90 , 3.30 , 3.70 , 4.10 close brace")[$\{ 2.50 , 2.90 , 3.30 , 3.70 , 4.10 \}$]
+ \$#math.equation(block: false, alt: "13.30")[$13.30$]


        
      
      
        If you order from Coldwater Creek, the shipping charges are given by the following table.


        #figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([Purchase #linebreak() amount #linebreak()], [Shipping #linebreak() charge #linebreak()]),
  [Up to \$#math.equation(block: false, alt: "25")[$25$]], [\$#math.equation(block: false, alt: "5.95")[$5.95$]],
  [\$#math.equation(block: false, alt: "25.01")[$25.01$] to \$#math.equation(block: false, alt: "50")[$50$]], [\$#math.equation(block: false, alt: "7.95")[$7.95$]],
  [\$#math.equation(block: false, alt: "50.01")[$50.01$] to \$#math.equation(block: false, alt: "75")[$75$]], [\$#math.equation(block: false, alt: "9.95")[$9.95$]],
  [\$#math.equation(block: false, alt: "75.01")[$75.01$] to \$#math.equation(block: false, alt: "100")[$100$]], [\$#math.equation(block: false, alt: "10.95")[$10.95$]],
))


        State the domain and range of #math.equation(block: false, alt: "S open parenthesis x close parenthesis")[$S ( x )$], the shipping charge as a function of the purchase amount, #math.equation(block: false, alt: "x")[$x$].


      
      
        The Bopp-Busch Tool and Die Company markets its products to individuals, to contractors, and to wholesale distributors. The company offers three different price structures for its toggle bolts. If you order #math.equation(block: false, alt: "20")[$20$] or fewer boxes, the price is \$#math.equation(block: false, alt: "2.50")[$2.50$] each. If you order more than #math.equation(block: false, alt: "20")[$20$] but no more than #math.equation(block: false, alt: "50")[$50$] boxes, the price is \$#math.equation(block: false, alt: "2.25")[$2.25$] each. If you order more than #math.equation(block: false, alt: "50")[$50$] boxes, the price is \$#math.equation(block: false, alt: "2.10")[$2.10$] each. State the domain and range of #math.equation(block: false, alt: "C open parenthesis x close parenthesis")[$C ( x )$], the cost of ordering #math.equation(block: false, alt: "x")[$x$] boxes of toggle bolts.


        
          Domain: nonnegative integers; The range includes all whole number multiples of #math.equation(block: false, alt: "2.50")[$2.50$] up to #math.equation(block: false, alt: "20 times 2.50 equals 50")[$20 × 2.50 = 50$], all integer multiples of #math.equation(block: false, alt: "2.25")[$2.25$] from #math.equation(block: false, alt: "21 times 2.25 equals 47.25")[$21 × 2.25 = 47.25$] to #math.equation(block: false, alt: "50 times 2.25 equals 112.50")[$50 × 2.25 = 112.50$] and all integer multiples of #math.equation(block: false, alt: "2.10")[$2.10$] from #math.equation(block: false, alt: "51 times 2.10 equals 107.10")[$51 × 2.10 = 107.10$] onwards: #math.equation(block: false, alt: "0 , 2.50 , 5.00 , 7.50 , and so on , 50")[$0 , 2.50 , 5.00 , 7.50 , … , 50$], #math.equation(block: false, alt: "47.25 , 49.50 , 51.75 , and so on , 112.50")[$47.25 , 49.50 , 51.75 , … , 112.50$], #math.equation(block: false, alt: "107.10 , 109.20 , 111.30 , and so on")[$107.10 , 109.20 , 111.30 , …$]


        
      
      
        The Java Stop uses paper cups at a rate of #math.equation(block: false, alt: "300")[$300$] per day. At opening on Tuesday morning Java Stop has on hand #math.equation(block: false, alt: "1200")[$1200$] paper cups. On Friday mornings Java Stop takes delivery of a week's worth of cups.


        + Write a piecewise function for the number of cups Java Stop has on hand for one week, starting Tuesday morning.
+ Graph the function.
+ State the domain and range of the function.


      
      For Problems 25–30, find the domain of each function algebraically. Then graph the function, and use the graph to help you find the range.


      
        + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the fraction 1 over open parenthesis x minus 4 close parenthesis squared")[$f ( x ) = display(frac(1, ( x − 4 )^(2)))$]
+ #math.equation(block: false, alt: "h open parenthesis x close parenthesis equals the fraction 1 over x squared minus 4")[$h ( x ) = display(frac(1, x^(2))) − 4$]


        
          + #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] domain: #math.equation(block: false, alt: "x not equal to 4")[$x ≠ 4$]; Range: #math.equation(block: false, alt: "open parenthesis 0 , ∞ close parenthesis")[$( 0 , upright(∞) )$]#figure(figph[horizontally shifted reciprocal-squared], alt: "horizontally shifted reciprocal-squared", caption: none)
+ #math.equation(block: false, alt: "h open parenthesis x close parenthesis")[$h ( x )$] domain: #math.equation(block: false, alt: "x not equal to 0")[$x ≠ 0$]; Range: #math.equation(block: false, alt: "open parenthesis minus 4 , ∞ close parenthesis")[$( − 4 , upright(∞) )$]#figure(figph[vertically shifted reciprocal-squared], alt: "vertically shifted reciprocal-squared", caption: none)


        
      
      
        + #math.equation(block: false, alt: "g open parenthesis t close parenthesis equals the fraction 1 over t plus 2")[$g ( t ) = display(frac(1, t)) + 2$]
+ #math.equation(block: false, alt: "F open parenthesis t close parenthesis equals the fraction 1 over t plus 2")[$F ( t ) = display(frac(1, t + 2))$]


      
      
        + #math.equation(block: false, alt: "G open parenthesis v close parenthesis equals v cubed plus 35")[$G ( v ) = v^(3) + 35$]
+ #math.equation(block: false, alt: "H open parenthesis v close parenthesis equals open parenthesis v plus 5 close parenthesis cubed")[$H ( v ) = ( v + 5 )^(3)$]


        
          + #math.equation(block: false, alt: "G open parenthesis v close parenthesis")[$G ( v )$] domain: all real numbers; Range: all real numbers#figure(figph[vertically shifted reciprocal], alt: "vertically shifted reciprocal", caption: none)
+ #math.equation(block: false, alt: "H open parenthesis v close parenthesis")[$H ( v )$] domain: all real numbers; Range: all real numbers#figure(figph[horizontally shifted reciprocal], alt: "horizontally shifted reciprocal", caption: none)


        
      
      
        + #math.equation(block: false, alt: "h open parenthesis n close parenthesis equals 3 plus open parenthesis n minus 1 close parenthesis squared")[$h ( n ) = 3 + ( n − 1 )^(2)$]
+ #math.equation(block: false, alt: "g open parenthesis n close parenthesis equals 3 minus open parenthesis n plus 1 close parenthesis squared")[$g ( n ) = 3 − ( n + 1 )^(2)$]


      
      
        + #math.equation(block: false, alt: "T open parenthesis z close parenthesis equals the square root of z minus 2")[$T ( z ) = sqrt(z − 2)$]
+ #math.equation(block: false, alt: "S open parenthesis z close parenthesis equals the square root of z minus 2")[$S ( z ) = sqrt(z) − 2$]


        
          + #math.equation(block: false, alt: "G open parenthesis v close parenthesis")[$G ( v )$] domain: #math.equation(block: false, alt: "open bracket 2 , ∞ close parenthesis")[$[ 2 , upright(∞) )$]; Range: #math.equation(block: false, alt: "open bracket 0 , ∞ close parenthesis")[$[ 0 , upright(∞) )$]#figure(figph[horizontally shifted root], alt: "horizontally shifted root", caption: none)
+ #math.equation(block: false, alt: "H open parenthesis v close parenthesis")[$H ( v )$] domain: #math.equation(block: false, alt: "open bracket 0 , ∞ close parenthesis")[$[ 0 , upright(∞) )$]; Range: #math.equation(block: false, alt: "open bracket minus 2 , ∞ close parenthesis")[$[ − 2 , upright(∞) )$]#figure(figph[vertically shifted root], alt: "vertically shifted root", caption: none)


        
      
      
        + #math.equation(block: false, alt: "Q open parenthesis x close parenthesis equals 4 minus vertical bar x vertical bar")[$Q ( x ) = 4 − | x |$]
+ #math.equation(block: false, alt: "P open parenthesis x close parenthesis equals vertical bar 4 minus x vertical bar")[$P ( x ) = | 4 − x |$]


      
      For Problems 31–38, decide whether the given value is in the range of the function. If so, find the domain value(s) that produce each range value.


      
        #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals 6 minus vertical bar 2 x plus 4 vertical bar")[$f ( x ) = 6 − | 2 x + 4 |$]


        + #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals 8")[$f ( x ) = 8$]
+ #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals minus 2")[$f ( x ) = − 2$]


        
          + Not in range
+ #math.equation(block: false, alt: "x equals minus 6")[$x = − 6$] or #math.equation(block: false, alt: "x equals 2")[$x = 2$]


        
      
      
        #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals open parenthesis x minus 5 close parenthesis cubed plus 1")[$g ( x ) = ( x − 5 )^(3) + 1$]


        + #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals 0")[$g ( x ) = 0$]
+ #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals minus 7")[$g ( x ) = − 7$]


      
      
        #math.equation(block: true, alt: "h open parenthesis t close parenthesis equals 4 plus 2 the cube root of t")[$h ( t ) = 4 + 2 root(3, t)$]


        + #math.equation(block: false, alt: "h open parenthesis t close parenthesis equals minus 4")[$h ( t ) = − 4$]
+ #math.equation(block: false, alt: "h open parenthesis t close parenthesis equals 0")[$h ( t ) = 0$]


        
          + #math.equation(block: false, alt: "t equals minus 64")[$t = − 64$]
+ #math.equation(block: false, alt: "t equals minus 8")[$t = − 8$]


        
      
      
        #math.equation(block: true, alt: "F open parenthesis t close parenthesis equals 12 plus 0.5 open parenthesis t minus 2 close parenthesis squared")[$F ( t ) = 12 + 0.5 ( t − 2 )^(2)$]


        + #math.equation(block: false, alt: "F open parenthesis t close parenthesis equals 10")[$F ( t ) = 10$]
+ #math.equation(block: false, alt: "F open parenthesis t close parenthesis equals 20")[$F ( t ) = 20$]


      
      
        #math.equation(block: true, alt: "G open parenthesis w close parenthesis equals 3 plus the fraction 2 over w minus 1")[$G ( w ) = 3 + display(frac(2, w − 1))$]


        + #math.equation(block: false, alt: "G open parenthesis w close parenthesis equals minus 1")[$G ( w ) = − 1$]
+ #math.equation(block: false, alt: "G open parenthesis w close parenthesis equals 3")[$G ( w ) = 3$]


        
          + #math.equation(block: false, alt: "w equals the fraction 1 over 2")[$w = display(frac(1, 2))$]
+ Not in range


        
      
      
        #math.equation(block: true, alt: "H open parenthesis n close parenthesis equals the fraction 4 over open parenthesis n plus 2 close parenthesis squared minus 5")[$H ( n ) = display(frac(4, ( n + 2 )^(2))) − 5$]


        + #math.equation(block: false, alt: "H open parenthesis n close parenthesis equals minus 6")[$H ( n ) = − 6$]
+ #math.equation(block: false, alt: "H open parenthesis n close parenthesis equals minus 1")[$H ( n ) = − 1$]


      
      
        #math.equation(block: true, alt: "Q open parenthesis h close parenthesis equals 2 plus the square root of h plus 5")[$Q ( h ) = 2 + sqrt(h + 5)$]


        + #math.equation(block: false, alt: "Q open parenthesis h close parenthesis equals 1")[$Q ( h ) = 1$]
+ #math.equation(block: false, alt: "Q open parenthesis h close parenthesis equals 5")[$Q ( h ) = 5$]


        
          + Not in range
+ #math.equation(block: false, alt: "h equals 4")[$h = 4$]


        
      
      
        #math.equation(block: true, alt: "P open parenthesis q close parenthesis equals 8 minus the square root of 4 minus q")[$P ( q ) = 8 − sqrt(4 − q)$]


        + #math.equation(block: false, alt: "P open parenthesis q close parenthesis equals 4")[$P ( q ) = 4$]
+ #math.equation(block: false, alt: "P open parenthesis q close parenthesis equals 12")[$P ( q ) = 12$]


      
      For Problems 39–50,


      + Use a graphing utility to graph each function on the given domain. Using a TRACE feature, adjust #strong[Ymin] and #strong[Ymax] until you can estimate the range of the function.
+ Verify your answer algebraically by evaluating the function. State the domain and range in interval notation.


      
        #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x squared minus 4 x ; minus 2 less than or equal to x less than or equal to 5")[$f ( x ) = x^(2) − 4 x ";" " " " " − 2 ≤ x ≤ 5$]


        
          Domain: #math.equation(block: false, alt: "open bracket minus 2 , 5 close bracket")[$[ − 2 , 5 ]$]; Range: #math.equation(block: false, alt: "open bracket minus 4 , 12 close bracket")[$[ − 4 , 12 ]$]


        
      
      
        #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals 6 x minus x squared ; minus 1 less than or equal to x less than or equal to 5")[$g ( x ) = 6 x − x^(2) ";" " " " " − 1 ≤ x ≤ 5$]


      
      
        #math.equation(block: true, alt: "g open parenthesis t close parenthesis equals minus t squared minus 2 t ; minus 5 less than or equal to t less than or equal to 3")[$g ( t ) = − t^(2) − 2 t ";" " " " " − 5 ≤ t ≤ 3$]


        
          Domain: #math.equation(block: false, alt: "open bracket minus 5 , 3 close bracket")[$[ − 5 , 3 ]$]; Range: #math.equation(block: false, alt: "open bracket minus 15 , 1 close bracket")[$[ − 15 , 1 ]$]


        
      
      
        #math.equation(block: true, alt: "f open parenthesis t close parenthesis equals minus t squared minus 4 t ; minus 6 less than or equal to t less than or equal to 2")[$f ( t ) = − t^(2) − 4 t ";" " " " " − 6 ≤ t ≤ 2$]


      
      
        #math.equation(block: true, alt: "h open parenthesis x close parenthesis equals x cubed minus 1 ; minus 2 less than or equal to x less than or equal to 2")[$h ( x ) = x^(3) − 1 ";" " " " " − 2 ≤ x ≤ 2$]


        
          Domain: #math.equation(block: false, alt: "open bracket minus 2 , 2 close bracket")[$[ − 2 , 2 ]$]; Range: #math.equation(block: false, alt: "open bracket minus 9 , 7 close bracket")[$[ − 9 , 7 ]$]


        
      
      
        #math.equation(block: true, alt: "q open parenthesis x close parenthesis equals x cubed plus 4 ; minus 3 less than or equal to x less than or equal to 2")[$q ( x ) = x^(3) + 4 ";" " " " " − 3 ≤ x ≤ 2$]


      
      
        #math.equation(block: true, alt: "F open parenthesis t close parenthesis equals the square root of 8 minus t ; minus 1 less than or equal to t less than or equal to 8")[$F ( t ) = sqrt(8 − t) ";" " " " " − 1 ≤ t ≤ 8$]


        
          Domain: #math.equation(block: false, alt: "open bracket minus 1 , 8 close bracket")[$[ − 1 , 8 ]$]; Range: #math.equation(block: false, alt: "open bracket 0 , 3 close bracket")[$[ 0 , 3 ]$]


        
      
      
        #math.equation(block: true, alt: "G open parenthesis t close parenthesis equals the square root of t plus 6 ; minus 6 less than or equal to t less than or equal to 3")[$G ( t ) = sqrt(t + 6) ";" " " " " − 6 ≤ t ≤ 3$]


      
      
        #math.equation(block: true, alt: "G open parenthesis x close parenthesis equals the fraction 1 over 3 minus x ; minus 1.25 less than or equal to x less than or equal to 2.75")[$G ( x ) = display(frac(1, 3 − x)) ";" " " " " − 1.25 ≤ x ≤ 2.75$]


        
          Domain: #math.equation(block: false, alt: "open bracket minus 1.25 , 2.75 close bracket")[$[ − 1.25 , 2.75 ]$]; Range: #math.equation(block: false, alt: "open bracket the fraction 4 over 17 , 4 close bracket")[$[ display(frac(4, 17)) , 4 ]$]


        
      
      
        #math.equation(block: true, alt: "H open parenthesis x close parenthesis equals the fraction 1 over x minus 1 ; minus 3.25 less than or equal to x less than or equal to minus 1.25")[$H ( x ) = display(frac(1, x − 1)) ";" " " " " − 3.25 ≤ x ≤ − 1.25$]


      
      
        #math.equation(block: true, alt: "G open parenthesis x close parenthesis equals the fraction 1 over 3 minus x ; 3 less than x less than or equal to 6")[$G ( x ) = display(frac(1, 3 − x)) ";" " " " " 3 < x ≤ 6$]


        
          Domain: #math.equation(block: false, alt: "open parenthesis 3 , 6 close bracket")[$( 3 , 6 ]$]; Range: #math.equation(block: false, alt: "open bracket minus ∞ , the fraction minus 1 over 3 close bracket")[$[ − upright(∞) , display(frac(− 1, 3)) ]$]


        
      
      
        #math.equation(block: true, alt: "H open parenthesis x close parenthesis equals the fraction 1 over x minus 1 ; 1 less than x less than or equal to 4")[$H ( x ) = display(frac(1, x − 1)) ";" " " " " 1 < x ≤ 4$]


      
      
        + Show that the graph of #math.equation(block: false, alt: "y equals the square root of 16 minus x squared")[$y = sqrt(16 − x^(2))$] is a semicircle.
+ State the domain and range of the function.
+ Graph the function in the window
        #math.equation(block: true, alt: "Xmin, equals minus 6, Xmax equals 6; Ymin, equals 0, Ymax equals 8")[$"Xmin" & = − 6 &  & "Xmax" = 6 \ "Ymin" & = 0 &  & "Ymax" = 8$] 
        In what way is the calculator’s graph misleading?


        
          (Hint: Write the equation in the form #math.equation(block: false, alt: "x squared plus y squared equals r squared")[$x^(2) + y^(2) = r^(2)$]. See Algebra Skills Refresher Facts from Geometry to review circles.)


        
        
          + Squaring both sides of the equation gives the equation of the circle centered on the origin with radius #math.equation(block: false, alt: "4")[$4$], but the points in the third and fourth quadrants are extraneous solutions introduced by squaring. (The original equation allowed only #math.equation(block: false, alt: "y greater than or equal to 0")[$y ≥ 0$].)
+ Domain: #math.equation(block: false, alt: "open bracket minus 4 , 4 close bracket")[$[ − 4 , 4 ]$]; Range: #math.equation(block: false, alt: "open bracket 0 , 4 close bracket")[$[ 0 , 4 ]$]
+ #figure(figph[GC graph], alt: "GC graph", caption: none)

 #linebreak() The calculator does not show the graph extending down to the #math.equation(block: false, alt: "x")[$x$]-axis.


        
      
      
        + For what values of #math.equation(block: false, alt: "x")[$x$] is the function #math.equation(block: false, alt: "y equals the fraction 2 x minus 8 over x minus 4")[$y = display(frac(2 x − 8, x − 4))$] undefined?
+ Graph the function in the standard window. In what way is the calculator's graph misleading?
+ Graph the function in the window
        #math.equation(block: true, alt: "Xmin, equals minus 9.4, Xmax equals 9.4; Ymin, equals minus 10, Ymax equals 10")[$"Xmin" & = − 9.4 &  & "Xmax" = 9.4 \ "Ymin" & = − 10 &  & "Ymax" = 10$] 
        State the domain and range of the function.


      
      In Problems 53–60, find the domain and range of each transformation of the given function.


      
        #math.equation(block: true, 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: "y equals f open parenthesis x minus 2 close parenthesis")[$y = f ( x − 2 )$]
+ #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis minus 2")[$y = f ( x ) − 2$]
+ #math.equation(block: false, alt: "y equals f open parenthesis x minus 3 close parenthesis minus 5")[$y = f ( x − 3 ) − 5$]


        
          + Domain: #math.equation(block: false, alt: "x not equal to 2")[$x ≠ 2$]; Range: #math.equation(block: false, alt: "open parenthesis 0 , ∞ close parenthesis")[$( 0 , upright(∞) )$]
+ Domain: #math.equation(block: false, alt: "x not equal to 0")[$x ≠ 0$]; Range: #math.equation(block: false, alt: "open parenthesis minus 2 , ∞ close parenthesis")[$( − 2 , upright(∞) )$]
+ Domain: #math.equation(block: false, alt: "x not equal to 3")[$x ≠ 3$]; Range: #math.equation(block: false, alt: "open parenthesis minus 5 , ∞ close parenthesis")[$( − 5 , upright(∞) )$]


        
      
      
        #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the square root of x")[$f ( x ) = sqrt(x)$]


        + #math.equation(block: false, alt: "y equals minus f open parenthesis x close parenthesis")[$y = − f ( x )$]
+ #math.equation(block: false, alt: "y equals 4 plus f open parenthesis x close parenthesis")[$y = 4 + f ( x )$]
+ #math.equation(block: false, alt: "y equals 4 minus f open parenthesis x close parenthesis")[$y = 4 − f ( x )$]


      
      
        #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals x squared")[$f ( x ) = x^(2)$]


        + #math.equation(block: false, alt: "y equals minus 2 f open parenthesis x close parenthesis")[$y = − 2 f ( x )$]
+ #math.equation(block: false, alt: "y equals 6 minus 2 f open parenthesis x close parenthesis")[$y = 6 − 2 f ( x )$]
+ #math.equation(block: false, alt: "y equals 6 minus 2 f open parenthesis x plus 3 close parenthesis")[$y = 6 − 2 f ( x + 3 )$]


        
          + Domain: all real numbers; Range: #math.equation(block: false, alt: "open parenthesis minus ∞ , 0 close parenthesis")[$( − upright(∞) , 0 )$]
+ Domain: all real numbers; Range: #math.equation(block: false, alt: "open parenthesis minus ∞ , 6 close bracket")[$( − upright(∞) , 6 ]$]
+ Domain: all real numbers; Range: #math.equation(block: false, alt: "open parenthesis minus ∞ , 6 close bracket")[$( − upright(∞) , 6 ]$]


        
      
      
        #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals the fraction 1 over x")[$f ( x ) = display(frac(1, x))$]


        + #math.equation(block: false, alt: "y equals 3 f open parenthesis x close parenthesis")[$y = 3 f ( x )$]
+ #math.equation(block: false, alt: "y equals 3 plus f open parenthesis x minus 1 close parenthesis")[$y = 3 + f ( x − 1 )$]
+ #math.equation(block: false, alt: "y equals 3 minus f open parenthesis x minus 1 close parenthesis")[$y = 3 − f ( x − 1 )$]


      
      
        The domain of #math.equation(block: false, alt: "f")[$f$] is #math.equation(block: false, alt: "open bracket 0 , 10 close bracket")[$[ 0 , 10 ]$] and the range is #math.equation(block: false, alt: "open bracket minus 2 , 2 close bracket")[$[ − 2 , 2 ]$].


        + #math.equation(block: false, alt: "y equals f open parenthesis x minus 3 close parenthesis")[$y = f ( x − 3 )$]
+ #math.equation(block: false, alt: "y equals 3 f open parenthesis x close parenthesis")[$y = 3 f ( x )$]
+ #math.equation(block: false, alt: "y equals 2 f open parenthesis x minus 5 close parenthesis")[$y = 2 f ( x − 5 )$]


        
          + Domain: #math.equation(block: false, alt: "open bracket 3 , 13 close bracket")[$[ 3 , 13 ]$]; Range: #math.equation(block: false, alt: "open bracket minus 2 , 2 close bracket")[$[ − 2 , 2 ]$]
+ Domain: #math.equation(block: false, alt: "open bracket 0 , 10 close bracket")[$[ 0 , 10 ]$]; Range: #math.equation(block: false, alt: "open bracket minus 6 , 6 close bracket")[$[ − 6 , 6 ]$]
+ Domain: #math.equation(block: false, alt: "open bracket 5 , 15 close bracket")[$[ 5 , 15 ]$]; Range: #math.equation(block: false, alt: "open bracket minus 4 , 4 close bracket")[$[ − 4 , 4 ]$]


        
      
      
        The domain of #math.equation(block: false, alt: "f")[$f$] is #math.equation(block: false, alt: "open bracket minus 4 , 4 close bracket")[$[ − 4 , 4 ]$] and the range is #math.equation(block: false, alt: "open bracket 3 , 10 close bracket")[$[ 3 , 10 ]$].


        + #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis plus 10")[$y = f ( x ) + 10$]
+ #math.equation(block: false, alt: "y equals f open parenthesis x plus 10 close parenthesis")[$y = f ( x + 10 )$]
+ #math.equation(block: false, alt: "y equals f open parenthesis x minus 1 close parenthesis plus 4")[$y = f ( x − 1 ) + 4$]


      
      
        The domain of #math.equation(block: false, alt: "f")[$f$] is #math.equation(block: false, alt: "open parenthesis 0 , plus ∞ close parenthesis")[$( 0 , + upright(∞) )$] and the range is #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$].


        + #math.equation(block: false, alt: "y equals 5 f open parenthesis x close parenthesis")[$y = 5 f ( x )$]
+ #math.equation(block: false, alt: "y equals 3 f open parenthesis x plus 2 close parenthesis")[$y = 3 f ( x + 2 )$]
+ #math.equation(block: false, alt: "y equals 2 f open parenthesis x minus 3 close parenthesis plus 2")[$y = 2 f ( x − 3 ) + 2$]


        
          + Domain: #math.equation(block: false, alt: "open parenthesis 0 , ∞ close parenthesis")[$( 0 , upright(∞) )$]; Range: #math.equation(block: false, alt: "open parenthesis 0 , 5 close parenthesis")[$( 0 , 5 )$]
+ Domain: #math.equation(block: false, alt: "open parenthesis minus 2 , ∞ close parenthesis")[$( − 2 , upright(∞) )$]; Range: #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$]
+ Domain: #math.equation(block: false, alt: "open parenthesis 3 , ∞ close parenthesis")[$( 3 , upright(∞) )$]; Range: #math.equation(block: false, alt: "open parenthesis 2 , 4 close parenthesis")[$( 2 , 4 )$]


        
      
      
        The domain of #math.equation(block: false, alt: "f")[$f$] is #math.equation(block: false, alt: "open parenthesis minus 1 , 1 close parenthesis")[$( − 1 , 1 )$] and the range is #math.equation(block: false, alt: "open parenthesis minus ∞ , 0 close parenthesis")[$( − upright(∞) , 0 )$].


        + #math.equation(block: false, alt: "y equals f open parenthesis x plus 1 close parenthesis")[$y = f ( x + 1 )$]
+ #math.equation(block: false, alt: "y equals 3 minus f open parenthesis x plus 1 close parenthesis")[$y = 3 − f ( x + 1 )$]
+ #math.equation(block: false, alt: "y equals 4 plus 2 f open parenthesis x minus 1 close parenthesis")[$y = 4 + 2 f ( x − 1 )$]


      
      In Problems 61–64, use a graphing calculator to explore some properties of the basic functions.


      
        + Graph #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 x cubed")[$g ( x ) = x^(3)$] on the domain #math.equation(block: false, alt: "open bracket 0 , 1 close bracket")[$[ 0 , 1 ]$] and state the range of each function. On the interval #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$], which is greater, #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] or #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]?
+ Graph #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 x cubed")[$g ( x ) = x^(3)$] on the domain #math.equation(block: false, alt: "open bracket 1 , 10 close bracket")[$[ 1 , 10 ]$] and state the range of each function. On the interval #math.equation(block: false, alt: "open parenthesis 1 , 100 close parenthesis")[$( 1 , 100 )$], which is greater, #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] or #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]?


        
          + #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]
+ #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]


        
      
      
        + Graph #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x")[$f ( x ) = sqrt(x)$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the cube root of x")[$g ( x ) = root(3, x)$] on the domain #math.equation(block: false, alt: "open bracket 0 , 1 close bracket")[$[ 0 , 1 ]$] and state the range of each function. On the interval #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$], which is greater, #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] or #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]?
+ Graph #math.equation(block: false, alt: "f open parenthesis x close parenthesis equals the square root of x")[$f ( x ) = sqrt(x)$] and #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the cube root of x")[$g ( x ) = root(3, x)$] on the domain #math.equation(block: false, alt: "open bracket 1 , 100 close bracket")[$[ 1 , 100 ]$] and state the range of each function. On the interval #math.equation(block: false, alt: "open parenthesis 1 , 100 close parenthesis")[$( 1 , 100 )$], which is greater, #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] or #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]?


      
      
        + Graph #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")[$g ( x ) = display(frac(1, x^(2)))$] on the domain #math.equation(block: false, alt: "open bracket 0.01 , 1 close bracket")[$[ 0.01 , 1 ]$] and state the range of each function. On the interval #math.equation(block: false, alt: "open parenthesis 0 , 1 close parenthesis")[$( 0 , 1 )$], which is greater, #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] or #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]?
+ Graph #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")[$g ( x ) = display(frac(1, x^(2)))$] on the domain #math.equation(block: false, alt: "open bracket 1 , 10 close bracket")[$[ 1 , 10 ]$] and state the range of each function. On the interval #math.equation(block: false, alt: "open parenthesis 1 , ∞ close parenthesis")[$( 1 , upright(∞) )$], which is greater, #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$] or #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]?


        
          + #math.equation(block: false, alt: "g open parenthesis x close parenthesis")[$g ( x )$]
+ #math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]


        
      
      
        + Graph #math.equation(block: false, alt: "F open parenthesis x close parenthesis equals vertical bar x cubed vertical bar")[$F ( x ) = | x^(3) |$] in the #strong[ZDecimal] window. How does the graph compare to the graph of #math.equation(block: false, alt: "y equals x cubed")[$y = x^(3)$]?
+ Graph #math.equation(block: false, alt: "G open parenthesis x close parenthesis equals vertical bar the fraction 1 over x vertical bar")[$G ( x ) = | display(frac(1, x)) |$] in the #strong[ZDecimal] window. How does the graph compare to the graph of #math.equation(block: false, alt: "y equals the fraction 1 over x")[$y = display(frac(1, x))$]?


      
      
        The number of hours of daylight on the summer solstice is a function of latitude in the northern hemisphere. Give the domain and range of the function.


        
          Domain: #math.equation(block: false, alt: "open bracket 0 degrees , 90 degrees close bracket")[$[ 0^(∘) , 90^(∘) ]$]; Range: #math.equation(block: false, alt: "open bracket 12 , 24 close bracket")[$[ 12 , 24 ]$]


        
      
      
        A semicircular window has a radius of 2 feet. The area of a sector of the window (a pie-shaped wedge) is a function of the angle at the center of the circle. Give the domain and range of this function.