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== 2.7#h(0.6em)Chapter Summary and Review


    
      
=== Key Concepts


      + We can solve equations of the form #math.equation(block: false, alt: "a open parenthesis p x plus q close parenthesis squared plus r equals 0")[$a ( p x + q )^(2) + r = 0$] by extraction of roots.
+ The formula for compound interest is #math.equation(block: false, alt: "A equals P open parenthesis 1 plus r close parenthesis to the power n")[$A = P ( 1 + r )^(n)$].
+ Simple nonlinear equations can be solved by undoing the operations on the variable.
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[]

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"))[
#emph[]

The absolute value has the following properties:

e\>

#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 situations can be modeled by one of eight basic functions:#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "y equals x")[$y = x$]], [#math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$]], [#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 fraction 1 over x")[$y = display(frac(1, x))$]], [#math.equation(block: false, alt: "y equals the fraction 1 over x squared")[$y = display(frac(1, x^(2)))$]], [#math.equation(block: false, alt: "y equals the square root of x")[$y = sqrt(x)$]], [#math.equation(block: false, alt: "y equals the cube root of x")[$y = root(3, x)$]],
))
+ Functions can be defined piecewise, with different formulas on different intervals.
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Transformations of Functions]

- The graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis plus k")[$y = f ( x ) + k$] is #strong[shifted vertically] compared to the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$].
- The graph of #math.equation(block: false, alt: "y equals f open parenthesis x plus h close parenthesis")[$y = f ( x + h )$] is #strong[shifted horizontally] compared to the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$].
- The graph of #math.equation(block: false, alt: "y equals a f open parenthesis x close parenthesis")[$y = a f ( x )$] is #strong[stretched or compressed vertically] compared to the graph of y = f (x).
- The graph of #math.equation(block: false, alt: "y equals minus f open parenthesis x close parenthesis")[$y = − f ( x )$] is #strong[reflected about the #math.equation(block: false, alt: "x")[$x$]-axis] compared to the graph of #math.equation(block: false, alt: "y equals f open parenthesis x close parenthesis")[$y = f ( x )$].
]
+ A nonlinear graph may be #strong[concave up] or #strong[concave down]. If a graph is concave up, its slope is increasing. If it is concave down, its slope is decreasing.
+ The absolute value is used to model distance: The distance between two points #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "a")[$a$] is given by #math.equation(block: false, alt: "vertical bar x minus a vertical bar")[$| x − a |$].
+ #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Absolute Value Equations and Inequalities]

- The equation #math.equation(block: false, alt: "vertical bar a x plus b vertical bar equals c open parenthesis c greater than 0 close parenthesis")[$| a x + b | = c " " " " ( c > 0 )$] is equivalent to
                	#math.equation(block: true, alt: "a x plus b equals c or a x plus b equals minus c")[$a x + b = c " or " a x + b = − c$]
- If the solutions of the equation #math.equation(block: false, alt: "vertical bar a x plus b vertical bar equals c")[$| a x + b | = c$] are #math.equation(block: false, alt: "r")[$r$] and #math.equation(block: false, alt: "s")[$s$], with #math.equation(block: false, alt: "r less than s")[$r < s$], then the solutions of #math.equation(block: false, alt: "vertical bar a x plus b vertical bar less than c")[$| a x + b | < c$] are #math.equation(block: false, alt: "r less than x less than s")[$r < x < s$].
- If the solutions of the equation #math.equation(block: false, alt: "vertical bar a x plus b vertical bar equals c")[$| a x + b | = c$] are #math.equation(block: false, alt: "r")[$r$] and #math.equation(block: false, alt: "s")[$s$], with #math.equation(block: false, alt: "r less than s")[$r < s$], then the solutions of #math.equation(block: false, alt: "vertical bar a x plus b vertical bar greater than c")[$| a x + b | > c$] are  #math.equation(block: false, alt: "x less than r")[$x < r$] or #math.equation(block: false, alt: "x greater than s")[$x > s$].
]
+ We can use absolute value notation to express error tolerances in measurements.
+ 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 input values of all points on the graph, and the range is the set of output 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 many applications, however, we may restrict the domain and range of a function to suit the situation at hand.


    
    
      
=== Chapter 2 Review Problems


      For Problems 1-4, solve by extraction of roots.


      
        #math.equation(block: true, alt: "open parenthesis 2 x minus 5 close parenthesis squared equals 9")[$( 2 x − 5 )^(2) = 9$]


        
          #math.equation(block: false, alt: "x equals 1")[$x = 1$] or #math.equation(block: false, alt: "x equals 4")[$x = 4$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis 7 x minus 1 close parenthesis squared equals 15")[$( 7 x − 1 )^(2) = 15$]


      
      
        #math.equation(block: true, alt: "6 open parenthesis the fraction w minus 1 over 3 close parenthesis squared minus 4 equals 2")[$6 attach(( display(frac(w − 1, 3)) ), t: 2) − 4 = 2$]


        
          #math.equation(block: false, alt: "w equals minus 2")[$w = − 2$] or #math.equation(block: false, alt: "w equals 4")[$w = 4$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis the fraction 2 p over 5 close parenthesis squared equals minus 3")[$attach(( display(frac(2 p, 5)) ), t: 2) = − 3$]


      
      For problems 5-6, solve the formula for the specified variable.


      
        #math.equation(block: false, alt: "A equals P open parenthesis 1 plus r close parenthesis squared")[$A = P ( 1 + r )^(2)$] , for #math.equation(block: false, alt: "r")[$r$]


        
          #math.equation(block: true, alt: "r equals minus 1 plus or minus the square root of the fraction A over P")[$r = − 1 ± sqrt(display(frac(A, P)))$]


        
      
      
        #math.equation(block: false, alt: "V equals the fraction 4 over 3 π r cubed")[$V = display(frac(4, 3)) π r^(3)$], for #math.equation(block: false, alt: "r")[$r$]


      
      
        Lewis invested \$#math.equation(block: false, alt: "2000")[$2000$] in an account that compounds interest annually. He made no deposits or withdrawals after that. Two years later, he closed the account, withdrawing \$#math.equation(block: false, alt: "2464.20")[$2464.20$]. What interest rate did Lewis earn?


        
          #math.equation(block: true, alt: "11 %")[$11 upright(%)$]


        
      
      
        Earl borrowed \$#math.equation(block: false, alt: "5500")[$5500$] from his uncle for two years with interest compounded annually. At the end of two years, he owed his uncle \$#math.equation(block: false, alt: "6474.74")[$6474.74$]. What was the interest rate
on the loan?


      
      For Problems 9-14, solve.


      
        #math.equation(block: true, alt: "the cube root of P minus 1 equals 0.1")[$root(3, P − 1) = 0.1$]


        
          #math.equation(block: true, alt: "P equals 1.001")[$P = 1.001$]


        
      
      
        #math.equation(block: true, alt: "the fraction 1 over 1 minus t equals the fraction 2 over 3")[$display(frac(1, 1 − t)) = display(frac(2, 3))$]


      
      
        #math.equation(block: true, alt: "the fraction 3 over the square root of m plus 7 equals the fraction 1 over 2")[$display(frac(3, sqrt(m + 7))) = display(frac(1, 2))$]


        
          #math.equation(block: true, alt: "m equals 29")[$m = 29$]


        
      
      
        #math.equation(block: true, alt: "15 equals 3 the square root of w plus 1")[$15 = 3 sqrt(w + 1)$]


      
      
        #math.equation(block: true, alt: "4 r cubed minus 8 equals 100")[$4 r^(3) − 8 = 100$]


        
          #math.equation(block: true, alt: "r equals 3")[$r = 3$]


        
      
      
        #math.equation(block: true, alt: "5 s squared plus 6 equals 3 s squared plus 31")[$5 s^(2) + 6 = 3 s^(2) + 31$]


      
      For Problems 15-16, use the Pythagorean theorem to write and solve an equation.


      
        A widescreen television measures #math.equation(block: false, alt: "96")[$96$] cm by #math.equation(block: false, alt: "54")[$54$] cm. How long is the diagonal?


        
          #math.equation(block: false, alt: "the square root of 12 , 132 approximately equals 110")[$sqrt(12 "," 132) ≈ 110$] cm


        
      
      
        A #math.equation(block: false, alt: "15")[$15$]-foot ladder leans to the top of a #math.equation(block: false, alt: "12")[$12$]-foot fence. How far is the foot of the ladder from the base of the fence?


      
      For Problems 17-20, simplify.


      
        #math.equation(block: true, alt: "vertical bar minus 18 vertical bar minus vertical bar 20 vertical bar")[$| − 18 | − | 20 |$]


        
          #math.equation(block: true, alt: "minus 2")[$− 2$]


        
      
      
        #math.equation(block: true, alt: "vertical bar minus 2 times open parenthesis 3 minus 18 close parenthesis vertical bar")[$| − 2 ⋅ ( 3 − 18 ) |$]


      
      
        #math.equation(block: true, alt: "vertical bar minus 2 times 3 minus 18 vertical bar")[$| − 2 ⋅ 3 − 18 |$]


        
          #math.equation(block: true, alt: "24")[$24$]


        
      
      
        #math.equation(block: true, alt: "minus 2 times vertical bar 3 minus 18 vertical bar")[$− 2 ⋅ | 3 − 18 |$]


      
      For Problems 21-24, use the graph to solve the equation or inequality.


      
        Refer to the graph of #math.equation(block: false, alt: "y equals vertical bar the fraction x over 2 minus 1 vertical bar")[$y = | display(frac(x, 2)) − 1 |$]


        #figure(figph[abs(x/2-1)], alt: "abs(x/2-1)", caption: none)


        + Solve #math.equation(block: false, alt: "vertical bar the fraction x over 2 minus 1 vertical bar equals 2")[$| display(frac(x, 2)) − 1 | = 2$]
+ Solve #math.equation(block: false, alt: "vertical bar the fraction x over 2 minus 1 vertical bar less than 2")[$| display(frac(x, 2)) − 1 | < 2$]
+ Solve #math.equation(block: false, alt: "vertical bar the fraction x over 2 minus 1 vertical bar greater than or equal to 2")[$| display(frac(x, 2)) − 1 | ≥ 2$]


        
          + #math.equation(block: false, alt: "x equals minus 2")[$x = − 2$] or #math.equation(block: false, alt: "x equals 6")[$x = 6$]
+ #math.equation(block: false, alt: "open parenthesis minus 2 , 6 close parenthesis")[$( − 2 , 6 )$]
+ #math.equation(block: false, alt: "open parenthesis minus ∞ , minus 2 close bracket union open bracket 6 , plus ∞ close parenthesis")[$( − upright(∞) , − 2 ] ∪ [ 6 , + upright(∞) )$]


        
      
      
        Refer to the graph of #math.equation(block: false, alt: "y equals the fraction minus x squared over 2 plus x plus 1")[$y = display(frac(− x^(2), 2)) + x + 1$]


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


        + Solve #math.equation(block: false, alt: "the fraction minus x squared over 2 plus x plus 1 equals minus 3")[$display(frac(− x^(2), 2)) + x + 1 = − 3$]
+ Solve #math.equation(block: false, alt: "the fraction minus x squared over 2 plus x plus 1 greater than or equal to minus 3")[$display(frac(− x^(2), 2)) + x + 1 ≥ − 3$]
+ Solve #math.equation(block: false, alt: "the fraction minus x squared over 2 plus x plus 1 less than or equal to minus 3")[$display(frac(− x^(2), 2)) + x + 1 ≤ − 3$]


      
      
        Refer to the graph of #math.equation(block: false, alt: "y equals the fraction 6 over x squared minus 3 x plus 3")[$y = display(frac(6, x^(2) − 3 x + 3))$]


        #figure(figph[abs(x/2-1)], alt: "abs(x/2-1)", caption: none)


        + Solve #math.equation(block: false, alt: "2 equals the fraction 6 over x squared minus 3 x plus 3")[$2 = display(frac(6, x^(2) − 3 x + 3))$]
+ Solve #math.equation(block: false, alt: "2 greater than the fraction 6 over x squared minus 3 x plus 3")[$2 > display(frac(6, x^(2) − 3 x + 3))$]
+ Solve #math.equation(block: false, alt: "2 less than the fraction 6 over x squared minus 3 x plus 3")[$2 < display(frac(6, x^(2) − 3 x + 3))$]


        
          + #math.equation(block: false, alt: "x equals 0")[$x = 0$] or #math.equation(block: false, alt: "x equals 3")[$x = 3$]
+ #math.equation(block: false, alt: "open parenthesis minus ∞ , 0 close parenthesis union open parenthesis 3 , ∞ close parenthesis")[$( − upright(∞) , 0 ) ∪ ( 3 , upright(∞) )$]
+ #math.equation(block: false, alt: "open parenthesis 0 , 3 close parenthesis")[$( 0 , 3 )$]


        
      
      
        Refer to the graph of #math.equation(block: false, alt: "y equals vertical bar x cubed plus 3 x squared plus 3 x plus 1 vertical bar")[$y = | x^(3) + 3 x^(2) + 3 x + 1 |$]


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


        + Solve #math.equation(block: false, alt: "8 equals vertical bar x cubed plus 3 x squared plus 3 x plus 1 vertical bar")[$8 = | x^(3) + 3 x^(2) + 3 x + 1 |$]
+ Solve #math.equation(block: false, alt: "8 greater than vertical bar x cubed plus 3 x squared plus 3 x plus 1 vertical bar")[$8 > | x^(3) + 3 x^(2) + 3 x + 1 |$]
+ Solve #math.equation(block: false, alt: "8 less than vertical bar x cubed plus 3 x squared plus 3 x plus 1 vertical bar")[$8 < | x^(3) + 3 x^(2) + 3 x + 1 |$]


      
      For Problems 25-30, graph the piecewise defined function.


      
        #math.equation(block: true, alt: "f open parenthesis x close parenthesis equals open brace x plus 1, if x less than or equal to 0; x squared, if x greater than 0")[$f ( x ) = \{ x + 1 & "if " x ≤ 0 \ x^(2) & "if " x > 0$]


        
          #figure(figph[piecewise defined function], alt: "piecewise defined function", caption: none)


        
      
      
        #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals open brace x minus 1, if x less than or equal to 1; x cubed, if x greater than 1")[$g ( x ) = \{ x − 1 & "if " x ≤ 1 \ x^(3) & "if " x > 1$]


      
      
        #math.equation(block: true, alt: "H open parenthesis x close parenthesis equals open brace x squared, if x less than or equal to 0; the square root of x, if x greater than 0")[$H ( x ) = \{ x^(2) & "if " x ≤ 0 \ sqrt(x) & "if " x > 0$]


        
          #figure(figph[piecewise defined function], alt: "piecewise defined function", caption: none)


        
      
      
        #math.equation(block: true, alt: "F open parenthesis x close parenthesis equals open brace vertical bar x vertical bar, if x less than or equal to 0; the fraction 1 over x, if x greater than 0")[$F ( x ) = \{ | x | & "if " x ≤ 0 \ display(frac(1, x)) & "if " x > 0$]


      
      
        #math.equation(block: true, alt: "S open parenthesis x close parenthesis equals open brace x cubed, if x less than or equal to 1; vertical bar x vertical bar, if x greater than 1")[$S ( x ) = \{ x^(3) & "if " x ≤ 1 \ | x | & "if " x > 1$]


        
          #figure(figph[piecewise defined function], alt: "piecewise defined function", caption: none)


        
      
      
        #math.equation(block: true, alt: "T open parenthesis x close parenthesis equals open brace the fraction 1 over x squared, if x less than 0; the square root of x, if x greater than or equal to 0")[$T ( x ) = \{ display(frac(1, x^(2))) & "if " x < 0 \ sqrt(x) & "if " x ≥ 0$]


      
      For Problems 31–38,


      + Describe each function as transformation of a basic function.
+ Sketch a graph of the basic function and the given function on the same axes.


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


        
          + #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$] shifted up #math.equation(block: false, alt: "2")[$2$] units
+ #figure(figph[shifted absolute value], alt: "shifted absolute value", caption: none)


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


      
      
        #math.equation(block: true, alt: "f open parenthesis s close parenthesis equals the square root of s plus 3")[$f ( s ) = sqrt(s) + 3$]


        
          + #math.equation(block: false, alt: "y equals the square root of x")[$y = sqrt(x)$] shifted up #math.equation(block: false, alt: "3")[$3$] units
+ #figure(figph[shifted square root], alt: "shifted square root", caption: none)


        
      
      
        #math.equation(block: true, alt: "g open parenthesis u close parenthesis equals the square root of u plus 2 minus 3")[$g ( u ) = sqrt(u + 2) − 3$]


      
      
        #math.equation(block: true, alt: "G open parenthesis t close parenthesis equals vertical bar t plus 2 vertical bar minus 3")[$G ( t ) = | t + 2 | − 3$]


        
          + #math.equation(block: false, alt: "y equals vertical bar x vertical bar")[$y = | x |$] shifted left #math.equation(block: false, alt: "2")[$2$] units and down #math.equation(block: false, alt: "3")[$3$] units
+ #figure(figph[shifted absolute value], alt: "shifted absolute value", caption: none)


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


      
      
        #math.equation(block: true, alt: "h open parenthesis s close parenthesis equals minus 2 the square root of s")[$h ( s ) = − 2 sqrt(s)$]


        
          + #math.equation(block: false, alt: "y equals the square root of x")[$y = sqrt(x)$] reflected across the horizontal axis and stretched vertically by a factor of #math.equation(block: false, alt: "2")[$2$]
+ #figure(figph[reflected and stretched square root], alt: "reflected and stretched square root", caption: none)


        
      
      
        #math.equation(block: true, alt: "H open parenthesis t close parenthesis equals the fraction 1 over 2 vertical bar s vertical bar")[$H ( t ) = display(frac(1, 2)) | s |$]


      
      In Problems 39–42, write a formula for each transformation of the given function.


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


        
          + #math.equation(block: false, alt: "y equals the fraction minus 3 over 2 f open parenthesis t close parenthesis")[$y = display(frac(− 3, 2)) f ( t )$]
+ #math.equation(block: false, alt: "y equals the fraction minus 3 over 2 f open parenthesis t close parenthesis plus 3")[$y = display(frac(− 3, 2)) f ( t ) + 3$]
+ #math.equation(block: false, alt: "y equals the fraction minus 3 over 2 f open parenthesis t plus 2 close parenthesis plus 3")[$y = display(frac(− 3, 2)) f ( t + 2 ) + 3$]


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


      
      
        #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "0")[$0$]], [#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: "f open parenthesis t close parenthesis")[$f ( t )$]], [#math.equation(block: false, alt: "243")[$243$]], [#math.equation(block: false, alt: "81")[$81$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "1")[$1$]],
))


        + #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#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: "6")[$6$]]),
  [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "243")[$243$]], [#math.equation(block: false, alt: "81")[$81$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "1")[$1$]],
))
+ #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#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: "6")[$6$]]),
  [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "minus 243")[$− 243$]], [#math.equation(block: false, alt: "minus 81")[$− 81$]], [#math.equation(block: false, alt: "minus 27")[$− 27$]], [#math.equation(block: false, alt: "minus 9")[$− 9$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]],
))
+ #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "t")[$t$]], [#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: "6")[$6$]]),
  [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "57")[$57$]], [#math.equation(block: false, alt: "219")[$219$]], [#math.equation(block: false, alt: "273")[$273$]], [#math.equation(block: false, alt: "291")[$291$]], [#math.equation(block: false, alt: "297")[$297$]], [#math.equation(block: false, alt: "299")[$299$]],
))


        
          + #math.equation(block: false, alt: "y equals f open parenthesis t minus 1 close parenthesis")[$y = f ( t − 1 )$]
+ #math.equation(block: false, alt: "y equals minus f open parenthesis t minus 1 close parenthesis")[$y = − f ( t − 1 )$]
+ #math.equation(block: false, alt: "y equals minus f open parenthesis t minus 1 close parenthesis plus 300")[$y = − f ( t − 1 ) + 300$]


        
      
      
        #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#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: "6")[$6$]]),
  [#math.equation(block: false, alt: "f open parenthesis x close parenthesis")[$f ( x )$]], [#math.equation(block: false, alt: "25")[$25$]], [#math.equation(block: false, alt: "24")[$24$]], [#math.equation(block: false, alt: "21")[$21$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "0")[$0$]],
))


        + #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#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: "y")[$y$]], [#math.equation(block: false, alt: "25")[$25$]], [#math.equation(block: false, alt: "24")[$24$]], [#math.equation(block: false, alt: "21")[$21$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "0")[$0$]],
))
+ #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#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: "y")[$y$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "48")[$48$]], [#math.equation(block: false, alt: "42")[$42$]], [#math.equation(block: false, alt: "32")[$32$]], [#math.equation(block: false, alt: "18")[$18$]], [#math.equation(block: false, alt: "0")[$0$]],
))
+ #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "0")[$0$]], [#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: "y")[$y$]], [#math.equation(block: false, alt: "70")[$70$]], [#math.equation(block: false, alt: "68")[$68$]], [#math.equation(block: false, alt: "62")[$62$]], [#math.equation(block: false, alt: "52")[$52$]], [#math.equation(block: false, alt: "38")[$38$]], [#math.equation(block: false, alt: "20")[$20$]],
))


      
      For Problems 43-44, give an equation for the function graphed.


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


        
          #math.equation(block: true, alt: "y equals open parenthesis x minus 2 close parenthesis squared minus 4")[$y = ( x − 2 )^(2) − 4$]


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


      
      Sketch graphs to illustrate the situations in Problems 45 and 46.


      
        Inga runs hot water into the bathtub until it is about half full. Because the water is too hot, she lets it sit for a while before getting into the tub. After several minutes of bathing, she gets out and drains the tub. Graph the water level in the bathtub as a function of time, from the moment Inga starts filling the tub until it is drained.


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


        
      
      
        David turns on the oven and it heats up steadily until the proper baking temperature is reached. The oven maintains that temperature during the time David bakes a pot roast. When he turns the oven off, David leaves the oven door open for a few minutes, and the temperature drops fairly rapidly during that time. After David closes the door, the temperature continues to drop, but at a much slower rate. Graph the temperature of the oven as a function of time, from the moment David first turns on the oven until shortly after David closes the door when the oven is cooling.


      
      For Problems 47-48, match each table with its graph.


      
        + #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "8")[$8$]]),
  [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "14")[$14$]], [#math.equation(block: false, alt: "21")[$21$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "43")[$43$]],
))
+ #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "40")[$40$]]),
  [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "52")[$52$]], [#math.equation(block: false, alt: "65")[$65$]], [#math.equation(block: false, alt: "75")[$75$]], [#math.equation(block: false, alt: "83")[$83$]],
))
+ #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "0")[$0$]], [#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: "y")[$y$]], [#math.equation(block: false, alt: "140")[$140$]], [#math.equation(block: false, alt: "190")[$190$]], [#math.equation(block: false, alt: "240")[$240$]], [#math.equation(block: false, alt: "290")[$290$]], [#math.equation(block: false, alt: "340")[$340$]],
))


        #figure(figph[three increasing curves], alt: "three increasing curves", caption: none)


        
          I (c), II (b), III (a)


        
      
      
        + #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0.1")[$0.1$]], [#math.equation(block: false, alt: "0.2")[$0.2$]], [#math.equation(block: false, alt: "0.3")[$0.3$]], [#math.equation(block: false, alt: "0.4")[$0.4$]]),
  [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "100")[$100$]], [#math.equation(block: false, alt: "95")[$95$]], [#math.equation(block: false, alt: "80")[$80$]], [#math.equation(block: false, alt: "55")[$55$]], [#math.equation(block: false, alt: "20")[$20$]],
))
+ #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "0")[$0$]], [#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: "y")[$y$]], [#math.equation(block: false, alt: "8.5")[$8.5$]], [#math.equation(block: false, alt: "7.1")[$7.1$]], [#math.equation(block: false, alt: "5.7")[$5.7$]], [#math.equation(block: false, alt: "4.3")[$4.3$]], [#math.equation(block: false, alt: "2.9")[$2.9$]],
))
+ #figure(table(
  columns: 6, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "40")[$40$]]),
  [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "37")[$37$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "15")[$15$]],
))


        #figure(figph[three decreasing curves], alt: "three decreasing curves", caption: none)


      
      Write and graph a piecewise function for Problems 49 and 50.


      
        The fluid level in a tank is a function of the number of days since the year began. The level was initially at #math.equation(block: false, alt: "60")[$60$] inches and rose an inch a day for #math.equation(block: false, alt: "10")[$10$] days, remained constant for the next #math.equation(block: false, alt: "20")[$20$] days, then dropped a half-inch each day for #math.equation(block: false, alt: "30")[$30$] days.


        
          #math.equation(block: true, alt: "g open parenthesis t close parenthesis equals open brace 60 plus t ,, 0 less than or equal to t less than 10; 70 ,, 10 less than or equal to t less than 30; 70 minus the fraction 1 over 2 open parenthesis t minus 30 close parenthesis ,, 30 less than or equal to t less than or equal to 60")[$g ( t ) = \{ 60 + t , & 0 ≤ t < 10 \ 70 , & 10 ≤ t < 30 \ 70 − frac(1, 2) ( t − 30 ) , & 30 ≤ t ≤ 60$]


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


        
      
      
        The temperature at different locations in a large room is a function of distance from the window. Within #math.equation(block: false, alt: "2")[$2$] feet of the window, the temperature is #math.equation(block: false, alt: "66 degrees")[$66^(∘)$] Fahrenheit, but the temperature rises by #math.equation(block: false, alt: "0.5 degrees")[$0.5^(∘)$] for each of the next #math.equation(block: false, alt: "10")[$10$] feet, then maintains the temperature at #math.equation(block: false, alt: "12")[$12$] feet for the rest of the room.


      
      For Problems 51-54, use absolute value notation to write the expression as an equation or inequality.


      
        #math.equation(block: false, alt: "x")[$x$] is four units from the origin.


        
          #math.equation(block: true, alt: "vertical bar x vertical bar equals 4")[$| x | = 4$]


        
      
      
        The distance from #math.equation(block: false, alt: "y")[$y$] to #math.equation(block: false, alt: "minus 5")[$− 5$] is three units.


      
      
        #math.equation(block: false, alt: "p")[$p$] is within four units of #math.equation(block: false, alt: "7")[$7$].


        
          #math.equation(block: true, alt: "vertical bar p minus 7 vertical bar less than 4")[$| p − 7 | < 4$]


        
      
      
        #math.equation(block: false, alt: "q")[$q$] is at least #math.equation(block: false, alt: "the fraction 3 over 10")[$display(frac(3, 10))$] unit from #math.equation(block: false, alt: "minus 4")[$− 4$].


      
      For Problems 55-64, solve.


      
        #math.equation(block: true, alt: "vertical bar 9 minus 5 t vertical bar equals 3")[$| 9 − 5 t | = 3$]


        
          #math.equation(block: false, alt: "t equals the fraction 6 over 5")[$t = display(frac(6, 5))$] or #math.equation(block: false, alt: "t equals the fraction 12 over 5")[$t = display(frac(12, 5))$]


        
      
      
        #math.equation(block: true, alt: "1 equals vertical bar 4 q minus 7 vertical bar")[$1 = | 4 q − 7 |$]


      
      
        #math.equation(block: true, alt: "minus 29 equals vertical bar 2 w plus 3 vertical bar")[$− 29 = | 2 w + 3 |$]


        
          No solutions


        
      
      
        #math.equation(block: true, alt: "vertical bar the fraction 8 n plus 3 over 5 vertical bar equals minus 11")[$| display(frac(8 n + 3, 5)) | = − 11$]


      
      
        #math.equation(block: true, alt: "1 equals vertical bar the fraction 7 minus 2 p over 5 vertical bar")[$1 = | display(frac(7 − 2 p, 5)) |$]


        
          #math.equation(block: false, alt: "p equals 1")[$p = 1$] or #math.equation(block: false, alt: "p equals 6")[$p = 6$]


        
      
      
        #math.equation(block: true, alt: "vertical bar 6 open parenthesis r minus 10 close parenthesis vertical bar equals 30")[$| 6 ( r − 10 ) | = 30$]


      
      
        #math.equation(block: true, alt: "vertical bar 3 x minus 2 vertical bar less than 4")[$| 3 x − 2 | < 4$]


        
          #math.equation(block: true, alt: "open parenthesis the fraction minus 2 over 3 , 2 close parenthesis")[$( display(frac(− 2, 3)) , 2 )$]


        
      
      
        #math.equation(block: true, alt: "vertical bar 2 x plus 0.3 vertical bar less than or equal to 0.5")[$| 2 x + 0.3 | ≤ 0.5$]


      
      
        #math.equation(block: true, alt: "vertical bar 3 y plus 1.2 vertical bar greater than or equal to 1.5")[$| 3 y + 1.2 | ≥ 1.5$]


        
          #math.equation(block: true, alt: "open parenthesis minus ∞ , minus 0.9 close bracket union open bracket 0.1 , ∞ close parenthesis")[$( − upright(∞) , − 0.9 ] ∪ [ 0.1 , upright(∞) )$]


        
      
      
        #math.equation(block: true, alt: "vertical bar 3 z plus the fraction 1 over 2 vertical bar greater than the fraction 1 over 3")[$| 3 z + display(frac(1, 2)) | > display(frac(1, 3))$]


      
      For Problems 65-66, express the error tolerance using absolute value.


      
        The height, #math.equation(block: false, alt: "H")[$H$], of a female trainee must be between #math.equation(block: false, alt: "56")[$56$] inches and #math.equation(block: false, alt: "75")[$75$] inches.


        
          #math.equation(block: true, alt: "vertical bar H minus 65.5 vertical bar less than 9.5")[$| H − 65.5 | < 9.5$]


        
      
      
        The time, #math.equation(block: false, alt: "t")[$t$], in freefall must be at least #math.equation(block: false, alt: "3.5")[$3.5$] seconds but no more than #math.equation(block: false, alt: "8.1")[$8.1$] seconds.


      
      For Problems 67-68, give an interval of possible values for the measurement.


      
        The mass, #math.equation(block: false, alt: "M")[$M$], of the sample must satisfy #math.equation(block: false, alt: "vertical bar M minus 2.1 vertical bar less than or equal to 0.05")[$| M − 2.1 | ≤ 0.05$].


        
          #math.equation(block: true, alt: "open bracket 2.05 , 2.15 close bracket")[$[ 2.05 , 2.15 ]$]


        
      
      
        The temperature, #math.equation(block: false, alt: "T")[$T$], of the refrigerator is specified by #math.equation(block: false, alt: "vertical bar T minus 4.0 vertical bar less than 0.5")[$| T − 4.0 | < 0.5$].


      
      In Problems 69 and 70,


      + Plot the points and sketch a smooth curve through them.
+ Use your graph to help you discover the equation that describes the function.


      
        #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")[$g ( x )$]]),
  [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "12")[$12$]],
  [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "8")[$8$]],
  [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "6")[$6$]],
  [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "4")[$4$]],
  [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "3")[$3$]],
  [#math.equation(block: false, alt: "12")[$12$]], [#math.equation(block: false, alt: "2")[$2$]],
))


        
          + #figure(figph[curve], alt: "curve", caption: none)
+ #math.equation(block: false, alt: "g open parenthesis x close parenthesis equals the fraction 24 over x")[$g ( x ) = display(frac(24, x))$]


        
      
      
        #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")[$F ( x )$]]),
  [#math.equation(block: false, alt: "minus 2")[$− 2$]], [#math.equation(block: false, alt: "8")[$8$]],
  [#math.equation(block: false, alt: "minus 1")[$− 1$]], [#math.equation(block: false, alt: "1")[$1$]],
  [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]],
  [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "minus 1")[$− 1$]],
  [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "minus 8")[$− 8$]],
  [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "minus 27")[$− 27$]],
))


      
      In Problems 71–76,


      + Use the graph to complete the table of values.
+ By finding a pattern in the table of values, write an equation for the graph.


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


        #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "8")[$8$]], [$#hide($000$)$], [#math.equation(block: false, alt: "16")[$16$]], [$#hide($000$)$]),
  [#math.equation(block: false, alt: "y")[$y$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [#math.equation(block: false, alt: "10")[$10$]], [$#hide($000$)$], [#math.equation(block: false, alt: "2")[$2$]],
))


        
          + #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "14")[$14$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "22")[$22$]]),
  [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "24")[$24$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "2")[$2$]],
))
+ #math.equation(block: false, alt: "y equals 24 minus x")[$y = 24 − x$]


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


        #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "10")[$10$]], [$#hide($000$)$], [#math.equation(block: false, alt: "14")[$14$]], [$#hide($000$)$]),
  [#math.equation(block: false, alt: "y")[$y$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [#math.equation(block: false, alt: "18")[$18$]], [$#hide($000$)$], [#math.equation(block: false, alt: "24")[$24$]],
))


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


        #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "0")[$0$]], [$#hide($000$)$], [#math.equation(block: false, alt: "4")[$4$]], [$#hide($000$)$], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "25")[$25$]]),
  [#math.equation(block: false, alt: "y")[$y$]], [$#hide($000$)$], [#math.equation(block: false, alt: "1")[$1$]], [$#hide($000$)$], [#math.equation(block: false, alt: "3")[$3$]], [$#hide($000$)$], [$#hide($000$)$],
))


        
          + #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "25")[$25$]]),
  [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "0")[$0$]], [#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: "y equals the square root of x")[$y = sqrt(x)$]


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


        #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [$#hide($000$)$], [#math.equation(block: false, alt: "0.5")[$0.5$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "1.5")[$1.5$]], [$#hide($000$)$], [#math.equation(block: false, alt: "4")[$4$]]),
  [#math.equation(block: false, alt: "y")[$y$]], [#math.equation(block: false, alt: "4")[$4$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$], [#math.equation(block: false, alt: "0.5")[$0.5$]], [$#hide($000$)$],
))


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


        #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [$#hide($000$)$], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]]),
  [#math.equation(block: false, alt: "y")[$y$]], [$#hide($000$)$], [$#hide($000$)$], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [$#hide($000$)$], [$#hide($000$)$], [$#hide($000$)$],
))


        
          + #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#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: "0")[$0$]], [#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: "5")[$5$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "0")[$0$]],
))
+ #math.equation(block: false, alt: "y equals x squared minus 4")[$y = x^(2) − 4$]


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


        #figure(table(
  columns: 7, align: left, inset: 6pt,
  table.header([#math.equation(block: false, alt: "x")[$x$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 2")[$− 2$]], [$#hide($000$)$], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [$#hide($000$)$]),
  [#math.equation(block: false, alt: "y")[$y$]], [$#hide($000$)$], [$#hide($000$)$], [#math.equation(block: false, alt: "8")[$8$]], [$#hide($000$)$], [$#hide($000$)$], [#math.equation(block: false, alt: "minus 7")[$− 7$]],
))


      
      For Problems 77-80, use a graphing calculator to graph the function on the given domain. Adjust #strong[Ymin] and #strong[Ymax] until you can determine the range of the function using the TRACE key. Then verify your answer algebraically by evaluating the function. State the domain and corresponding range in interval notation.


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


        
          Domain: #math.equation(block: false, alt: "open bracket minus 2 , 4 close bracket")[$[ − 2 , 4 ]$]; Range: #math.equation(block: false, alt: "open bracket minus 10 , minus 4 close bracket")[$[ − 10 , − 4 ]$]


        
      
      
        #math.equation(block: true, alt: "g open parenthesis x close parenthesis equals the square root of s minus 2")[$g ( x ) = sqrt(s − 2)$]; #math.equation(block: true, alt: "2 less than or equal to s less than or equal to 6")[$" " " " 2 ≤ s ≤ 6$]


      
      
        #math.equation(block: true, alt: "F open parenthesis x close parenthesis equals the fraction 1 over x plus 2")[$F ( x ) = display(frac(1, x + 2))$]; #math.equation(block: true, alt: "minus 2 less than x less than or equal to 4")[$" " " " − 2 < x ≤ 4$]


        
          Domain: #math.equation(block: false, alt: "open parenthesis minus 2 , 4 close bracket")[$( − 2 , 4 ]$]; Range: #math.equation(block: false, alt: "open bracket the fraction 1 over 6 , ∞ close parenthesis")[$[ display(frac(1, 6)) , upright(∞) )$]


        
      
      
        #math.equation(block: true, alt: "H open parenthesis x close parenthesis equals the fraction 1 over 2 minus x")[$H ( x ) = display(frac(1, 2 − x))$]; #math.equation(block: true, alt: "minus 4 less than or equal to x less than 2")[$" " " " − 4 ≤ x < 2$]