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13.2.1: Chapter 1

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    117276
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    Try It

    1.1 Functions and Function Notation

    1.

    1. ⓐ yes
    2. ⓑ yes. (Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)

    2.

    w=f(d)w=f(d)

    3.

    yes

    4.

    g(5)=1g(5)=1

    5.

    m=8m=8

    6.

    y=f(x)=x√32y=f(x)=x32

    7.

    g(1)=8g(1)=8

    8.

    x=0x=0 or x=2x=2

    9.

    1. ⓐ yes, because each bank account has a single balance at any given time
    2. ⓑ no, because several bank account numbers may have the same balance
    3. ⓒ no, because the same output may correspond to more than one input.

    10.

    1. ⓐ Yes, letter grade is a function of percent grade;
    2. ⓑ No, it is not one-to-one. There are 100 different percent numbers we could get but only about five possible letter grades, so there cannot be only one percent number that corresponds to each letter grade.

    11.

    yes

    12.

    No, because it does not pass the horizontal line test.

    1.2 Domain and Range

    1.

    {−5,0,5,10,15}{−5,0,5,10,15}

    2.

    (−∞,∞)(−∞,∞)

    3.

    (−∞,12)∪(12,∞)(−∞,12)∪(12,∞)

    4.

    [−52,∞)[ −52,∞ )

    5.

    1. ⓐ values that are less than or equal to –2, or values that are greater than or equal to –1 and less than 3;
    2. ⓑ {x|x≤−2or−1≤x<3}{ x|x≤−2or−1≤x<3 } ;
    3. ⓒ (−∞,−2]∪[−1,3)(−∞,−2]∪[−1,3)

    6.

    domain =[1950,2002] range = [47,000,000,89,000,000]

    7.

    domain: (−∞,2];( −∞,2 ]; range: (−∞,0]( −∞,0 ]

    8.

    Graph of f(x).

    1.3 Rates of Change and Behavior of Graphs

    1.

    $2.84−$2.315 years=$0.535 years=$0.106$2.84−$2.315 years=$0.535 years=$0.106 per year.

    2.

    1212

    3.

    a+7a+7

    4.

    The local maximum appears to occur at (−1,28),(−1,28), and the local minimum occurs at (5,−80).(5,−80). The function is increasing on (−∞,−1)∪(5,∞)(−∞,−1)∪(5,∞) and decreasing on (−1,5).(−1,5).

    Graph of a polynomial with a local maximum at (-1, 28) and local minimum at (5, -80).

    1.4 Composition of Functions

    1.

    (fg)(x)=f(x)g(x)=(x−1)(x2−1)=x3−x2−x+1(f−g)(x)=f(x)−g(x)=(x−1)−(x2−1)=x−x2(fg)(x)=f(x)g(x)=(x−1)(x2−1)=x3−x2−x+1(f−g)(x)=f(x)−g(x)=(x−1)−(x2−1)=x−x2

    No, the functions are not the same.

    2.

    A gravitational force is still a force, so a(G(r))a(G(r)) makes sense as the acceleration of a planet at a distance r from the Sun (due to gravity), but G(a(F))G(a(F)) does not make sense.

    3.

    f(g(1))=f(3)=3f(g(1))=f(3)=3 and g(f(4))=g(1)=3g(f(4))=g(1)=3

    4.

    g(f(2))=g(5)=3g(f(2))=g(5)=3

    5.

    1. ⓐ 8
    2. ⓑ 20

    6.

    [−4,0)∪(0,∞)[ −4,0 )∪(0,∞)

    7.

    Possible answer:

    g(x)=4+x2−−−−−√g(x)=4+x2
    h(x)=43−xh(x)=43−x
    f=h∘gf=h∘g

    1.5 Transformation of Functions

    1.

    b(t)=h(t)+10=−4.9t2+30t+10b(t)=h(t)+10=−4.9t2+30t+10

    2.

    The graphs of f(x)f(x) and g(x)g(x) are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units.

    Graph of a square root function and a horizontally shift square foot function.3.

    Graph of h(x)=|x-2|+4.

    4.

    g(x)=1x−1+1g(x)=1x-1+1

    5.

    1. Graph of a vertically reflected absolute function.
    2. Graph of an absolute function translated one unit left.

    6.

    1. g(x)=−f(x)g(x)=−f(x)

      xx -2 0 2 4
      g(x)g(x) −5−5 −10−10 −15−15 −20−20
    2. h(x)=f(−x)h(x)=f(−x)

      xx -2 0 2 4
      h(x)h(x) 15 10 5 unknown

    7.Graph of x^2 and its reflections.

    Notice: g(x)=f(−x)g(x)=f(−x) looks the same as f(x)f(x).

    8.

    even

    9.

    xx 2 4 6 8
    g(x)g(x) 9 12 15 0

    10.

    g(x)=3x−2g(x)=3x-2

    11.

    g(x)=f(13x)g(x)=f(13x) so using the square root function we get g(x)=13x−−−√g(x)=13x

    1.6 Absolute Value Functions

    1.

    |x−2|≤3| x−2 |≤3

    2.

    using the variable pp for passing, |p−80|≤20| p−80 |≤20

    3.

    f(x)=−|x+2|+3f(x)=−| x+2 |+3

    4.

    x=−1x=−1 or x=2x=2

    5.

    f(0)=1,f(0)=1, so the graph intersects the vertical axis at (0,1).(0,1). f(x)=0f(x)=0 when x=−5x=−5 and x=1x=1 so the graph intersects the horizontal axis at (−5,0)(−5,0) and (1,0).(1,0).

    6.

    −8≤x≤4-8≤x≤4

    7.

    k≤1k≤1 or k≥7;k≥7; in interval notation, this would be (−∞,1]∪[7,∞)(−∞,1]∪[7,∞)

    1.7 Inverse Functions

    1.

    h(2)=6h(2)=6

    2.

    No

    3.

    Yes

    4.

    The domain of function f−1f−1 is (−∞,−2)(−∞,−2) and the range of function f−1f−1 is (1,∞).(1,∞).

    5.

    1. f(60)=50.f(60)=50. In 60 minutes, 50 miles are traveled.
    2. f−1(60)=70.f−1(60)=70. To travel 60 miles, it will take 70 minutes.

    6.

    a. 3; b. 5.6

    7.

    x=3y+5x=3y+5

    8.

    f−1(x)=(2−x)2;domainoff:[0,∞);domainoff−1:(−∞,2]f−1(x)=(2−x)2;domainoff:[ 0,∞ );domainoff−1:( −∞,2 ]

    9.

    Graph of f(x) and f^(-1)(x).

    1.1 Section Exercises

    1.

    A relation is a set of ordered pairs. A function is a special kind of relation in which no two ordered pairs have the same first coordinate.

    3.

    When a vertical line intersects the graph of a relation more than once, that indicates that for that input there is more than one output. At any particular input value, there can be only one output if the relation is to be a function.

    5.

    When a horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than one input. A function is one-to-one if each output corresponds to only one input.

    7.

    function

    9.

    function

    11.

    function

    13.

    function

    15.

    function

    17.

    function

    19.

    function

    21.

    function

    23.

    function

    25.

    not a function

    27.

    f(−3)=−11;f(−3)=−11;
    f(2)=−1;f(2)=−1;
    f(−a)=−2a−5;f(−a)=−2a−5;
    −f(a)=−2a+5;−f(a)=−2a+5;
    f(a+h)=2a+2h−5f(a+h)=2a+2h−5

    29.

    f(−3)=5–√+5;f(−3)=5+5;
    f(2)=5;f(2)=5;
    f(−a)=2+a−−−−√+5;f(−a)=2+a+5;
    −f(a)=−2−a−−−−√−5;−f(a)=−2−a−5;
    f(a+h)=2−a−h−−−−−−−−√+5f(a+h)=2−a−h+5

    31.

    f(−3)=2;f(−3)=2; f(2)=1−3=−2;f(2)=1−3=−2;
    f(−a)=|−a−1|−|−a+1|;f(−a)=|−a−1|−|−a+1|;
    −f(a)=−|a−1|+|a+1|;−f(a)=−|a−1|+|a+1|;
    f(a+h)=|a+h−1|−|a+h+1|f(a+h)=|a+h−1|−|a+h+1|

    33.

    g(x)−g(a)x−a=x+a+2,x≠ag(x)−g(a)x−a=x+a+2,x≠a

    35.

    1. ⓐ f(−2)=14;f(−2)=14;
    2. ⓑ x=3x=3

    37.

    1. ⓐ f(5)=10;f(5)=10;
    2. ⓑ x=−1x=−1 or x=4x=4

    39.

    1. ⓐ f(t)=6−23t;f(t)=6−23t;
    2. ⓑ f(−3)=8;f(−3)=8;
    3. ⓒ t=6t=6

    41.

    not a function

    43.

    function

    45.

    function

    47.

    function

    49.

    function

    51.

    function

    53.

    1. ⓐ f(0)=1;f(0)=1;
    2. ⓑ f(x)=−3,x=−2f(x)=−3,x=−2 or x=2x=2

    55.

    not a function so it is also not a one-to-one function

    57.

    one-to-one function

    59.

    function, but not one-to-one

    61.

    function

    63.

    function

    65.

    not a function

    67.

    f(x)=1,x=2f(x)=1,x=2

    69.

    f(−2)=14;f(−1)=11;f(0)=8;f(1)=5;f(2)=2f(−2)=14;f(−1)=11;f(0)=8;f(1)=5;f(2)=2

    71.

    f(−2)=4;  f(−1)=4.414;f(0)=4.732;f(1)=5;f(2)=5.236f(−2)=4;  f(−1)=4.414;f(0)=4.732;f(1)=5;f(2)=5.236

    73.

    f(−2)=19;f(−1)=13;f(0)=1;f(1)=3;f(2)=9f(−2)=19;f(−1)=13;f(0)=1;f(1)=3;f(2)=9

    75.

    20

    77.

    [0, 100][0, 100]

    Graph of a parabola.79.

    [−0.001, 0.001][−0.001, 0.001]

    Graph of a parabola.81.

    [−1,000,000, 1,000,000][−1,000,000, 1,000,000]

    Graph of a cubic function.83.

    [0, 10][0, 10]

    Graph of a square root function.85.

    [−0.1,0.1][−0.1,0.1]

    Graph of a square root function.87.

    [−100, 100][−100, 100]

    Graph of a cubic root function.89.

    1. ⓐ g(5000)=50;g(5000)=50;
    2. ⓑ The number of cubic yards of dirt required for a garden of 100 square feet is 1.

    91.

    1. ⓐ The height of a rocket above ground after 1 second is 200 ft.
    2. ⓑ the height of a rocket above ground after 2 seconds is 350 ft.

    1.2 Section Exercises

    1.

    The domain of a function depends upon what values of the independent variable make the function undefined or imaginary.

    3.

    There is no restriction on xx for f(x)=x−−√3f(x)=x3 because you can take the cube root of any real number. So the domain is all real numbers, (−∞,∞).(−∞,∞). When dealing with the set of real numbers, you cannot take the square root of negative numbers. So xx-values are restricted for f(x)=x−−√f(x)=x to nonnegative numbers and the domain is [0,∞).[0,∞).

    5.

    Graph each formula of the piecewise function over its corresponding domain. Use the same scale for the xx-axis and yy-axis for each graph. Indicate inclusive endpoints with a solid circle and exclusive endpoints with an open circle. Use an arrow to indicate −∞−∞ or ∞.∞. Combine the graphs to find the graph of the piecewise function.

    7.

    (−∞,∞)(−∞,∞)

    9.

    (−∞,3](−∞,3]

    11.

    (−∞,∞)(−∞,∞)

    13.

    (−∞,∞)(−∞,∞)

    15.

    (−∞,−12)∪(−12,∞)(−∞,−12)∪(−12,∞)

    17.

    (−∞,−11)∪(−11,2)∪(2,∞)(−∞,−11)∪(−11,2)∪(2,∞)

    19.

    (−∞,−3)∪(−3,5)∪(5,∞)(−∞,−3)∪(−3,5)∪(5,∞)

    21.

    (−∞,5)(−∞,5)

    23.

    [6,∞)[6,∞)

    25.

    (−∞,−9)∪(−9,9)∪(9,∞)(−∞,−9)∪(−9,9)∪(9,∞)

    27.

    domain: (2,8],(2,8], range [6,8)[6,8)

    29.

    domain: [−4, 4],[−4, 4], range: [0, 2][0, 2]

    31.

    domain: [−5,3),[−5,3), range: [0,2][ 0,2 ]

    33.

    domain: (−∞,1],(−∞,1], range: [0,∞)[0,∞)

    35.

    domain: [−6,−16]∪[16,6];[ −6,−16 ]∪[ 16,6 ]; range: [−6,−16]∪[16,6][ −6,−16 ]∪[ 16,6 ]

    37.

    domain: [−3,∞);[−3,∞); range: [0,∞)[0,∞)

    39.

    domain: (−∞,∞)(−∞,∞)

    Graph of f(x).41.

    domain: (−∞,∞)(−∞,∞)

    Graph of f(x).43.

    domain: (−∞,∞)(−∞,∞)

    Graph of f(x).45.

    domain: (−∞,∞)(−∞,∞)

    Graph of f(x).47.

    f(−3)=1;f(−2)=0;f(−1)=0;f(0)=0f(−3)=1;f(−2)=0;f(−1)=0;f(0)=0

    49.

    f(−1)=−4;f(0)=6;f(2)=20;f(4)=34f(−1)=−4;f(0)=6;f(2)=20;f(4)=34

    51.

    f(−1)=−5;f(0)=3;f(2)=3;f(4)=16f(−1)=−5;f(0)=3;f(2)=3;f(4)=16

    53.

    domain: (−∞,1)∪(1,∞)(−∞,1)∪(1,∞)

    55.Graph of the equation from [-0.5, -0.1].

    window: [−0.5,−0.1];[−0.5,−0.1]; range: [4,100][4,100]

    Graph of the equation from [0.1, 0.5].

    window: [0.1,0.5];[0.1,0.5]; range: [4,100][4,100]

    57.

    [0,8][0,8]

    59.

    Many answers. One function is f(x)=1x−2√.f(x)=1x−2.

    1.3 Section Exercises

    1.

    Yes, the average rate of change of all linear functions is constant.

    3.

    The absolute maximum and minimum relate to the entire graph, whereas the local extrema relate only to a specific region around an open interval.

    5.

    4(b+1)4(b+1)

    7.

    3

    9.

    4x+2h4x+2h

    11.

    −113(13+h)−113(13+h)

    13.

    3h2+9h+93h2+9h+9

    15.

    4x+2h−34x+2h−3

    17.

    4343

    19.

    increasing on (−∞,−2.5)∪(1,∞),(−∞,−2.5)∪(1,∞), decreasing on (−2.5,1)(−2.5,1)

    21.

    increasing on (−∞,1)∪(3,4),(−∞,1)∪(3,4), decreasing on (1,3)∪(4,∞)(1,3)∪(4,∞)

    23.

    local maximum: (−3,50),(−3,50), local minimum: (3,−50)(3,−50)

    25.

    absolute maximum at approximately (7,150),(7,150), absolute minimum at approximately (−7.5,−220)(−7.5,−220)

    27.

    a. –3000; b. –1250

    29.

    -4

    31.

    27

    33.

    –0.167

    35.

    Local minimum at (3,−22),(3,−22), decreasing on (−∞,3),(−∞,3), increasing on (3,∞)(3,∞)

    37.

    Local minimum at (−2,−2),(−2,−2), decreasing on (−3,−2),(−3,−2), increasing on (−2,∞)(−2,∞)

    39.

    Local maximum at (−0.5,6),(−0.5,6), local minima at (−3.25,−47)(−3.25,−47) and (2.1,−32),(2.1,−32), decreasing on (−∞,−3.25)(−∞,−3.25) and (−0.5,2.1),(−0.5,2.1), increasing on (−3.25,−0.5)(−3.25,−0.5) and (2.1,∞)(2.1,∞)

    41.

    A

    43.

    b=5b=5

    45.

    2.7 gallons per minute

    47.

    approximately –0.6 milligrams per day

    1.4 Section Exercises

    1.

    Find the numbers that make the function in the denominator gg equal to zero, and check for any other domain restrictions on ff and g,g, such as an even-indexed root or zeros in the denominator.

    3.

    Yes. Sample answer: Let f(x)=x+1 and g(x)=x−1.f(x)=x+1 and g(x)=x−1. Then f(g(x))=f(x−1)=(x−1)+1=xf(g(x))=f(x−1)=(x−1)+1=x and g(f(x))=g(x+1)=(x+1)−1=x.g(f(x))=g(x+1)=(x+1)−1=x. So f∘g=g∘f.f∘g=g∘f.

    5.

    (f+g)(x)=2x+6,(f+g)(x)=2x+6, domain: (−∞,∞)(−∞,∞)

    (f−g)(x)=2x2+2x−6,(f−g)(x)=2x2+2x−6, domain: (−∞,∞)(−∞,∞)

    (fg)(x)=−x4−2x3+6x2+12x,(fg)(x)=−x4−2x3+6x2+12x, domain: (−∞,∞)(−∞,∞)

    (fg)(x)=x2+2x6−x2,(fg)(x)=x2+2x6−x2, domain: (−∞,−6–√)∪(−6–√,6–√)∪(6–√,∞)(−∞,−6)∪(−6,6)∪(6,∞)

    7.

    (f+g)(x)=4x3+8x2+12x,(f+g)(x)=4x3+8x2+12x, domain: (−∞,0)∪(0,∞)(−∞,0)∪(0,∞)

    (f−g)(x)=4x3+8x2−12x,(f−g)(x)=4x3+8x2−12x, domain: (−∞,0)∪(0,∞)(−∞,0)∪(0,∞)

    (fg)(x)=x+2,(fg)(x)=x+2, domain: (−∞,0)∪(0,∞)(−∞,0)∪(0,∞)

    (fg)(x)=4x3+8x2,(fg)(x)=4x3+8x2, domain: (−∞,0)∪(0,∞)(−∞,0)∪(0,∞)

    9.

    (f+g)(x)=3x2+x−5−−−−√,(f+g)(x)=3x2+x−5, domain: [5,∞)[5,∞)

    (f−g)(x)=3x2−x−5−−−−√,(f−g)(x)=3x2−x−5, domain: [5,∞)[5,∞)

    (fg)(x)=3x2x−5−−−−√,(fg)(x)=3x2x−5, domain: [5,∞)[5,∞)

    (fg)(x)=3x2x−5√,(fg)(x)=3x2x−5, domain: (5,∞)(5,∞)

    11.

    1. ⓐ 3
    2. ⓑ f(g(x))=2(3x−5)2+1;f(g(x))=2(3x−5)2+1;
    3. ⓒ g(f)(x))=6x2−2;g(f)(x))=6x2−2;
    4. ⓓ (g∘g)(x)=3(3x−5)−5=9x−20;(g∘g)(x)=3(3x−5)−5=9x−20;
    5. ⓔ (f∘f)(−2)=163(f∘f)(−2)=163

    13.

    f(g(x))=x2+3−−−−−√+2,g(f(x))=x+4x−−√+7f(g(x))=x2+3+2,g(f(x))=x+4x+7

    15.

    f(g(x))=x+1x3−−−√3=x+1√3x,g(f(x))=x√3+1xf(g(x))=x+1x33=x+13x,g(f(x))=x3+1x

    17.

    (f∘g)(x)=12x+4−4=x2,(g∘f)(x)=2x−4(f∘g)(x)=12x+4−4=x2,(g∘f)(x)=2x−4

    19.

    f(g(h(x)))=(1x+3)2+1f(g(h(x)))=(1x+3)2+1

    21.

    • ⓐ Text (g∘f)(x)=−32−4x√;(g∘f)(x)=−32−4x;
    • ⓑ(−∞,12)(−∞,12)

    23.

    1. ⓐ (0,2)∪(2,∞);(0,2)∪(2,∞);
    2. ⓑ (−∞,−2)∪(2,∞);(−∞,−2)∪(2,∞); c. (0,∞)(0,∞)

    25.

    (1,∞)(1,∞)

    27.

    sample: f(x)=x3g(x)=x−5f(x)=x3g(x)=x−5

    29.

    sample: f(x)=4xg(x)=(x+2)2f(x)=4xg(x)=(x+2)2

    31.

    sample: f(x)=x−−√3g(x)=12x−3f(x)=x3g(x)=12x−3

    33.

    sample: f(x)=x−−√4g(x)=3x−2x+5f(x)=x4g(x)=3x−2x+5

    35.

    sample: f(x)=x−−√f(x)=x
    g(x)=2x+6g(x)=2x+6

    37.

    sample: f(x)=x−−√3f(x)=x3
    g(x)=(x−1)g(x)=(x−1)

    39.

    sample: f(x)=x3f(x)=x3
    g(x)=1x−2g(x)=1x−2

    41.

    sample: f(x)=x−−√f(x)=x
    g(x)=2x−13x+4g(x)=2x−13x+4

    43.

    2

    45.

    5

    47.

    4

    49.

    0

    51.

    2

    53.

    1

    55.

    4

    57.

    4

    59.

    9

    61.

    4

    63.

    2

    65.

    3

    67.

    11

    69.

    0

    71.

    7

    73.

    f(g(0))=27,g(f(0))=−94f(g(0))=27,g(f(0))=−94

    75.

    f(g(0))=15,g(f(0))=5f(g(0))=15,g(f(0))=5

    77.

    18x2+60x+5118x2+60x+51

    79.

    g∘g(x)=9x+20g∘g(x)=9x+20

    81.

    2

    83.

    (−∞,∞)(−∞,∞)

    85.

    False

    87.

    (f∘g)(6)=6(f∘g)(6)=6; (g∘f)(6)=6(g∘f)(6)=6

    89.

    (f∘g)(11)=11,(g∘f)(11)=11(f∘g)(11)=11,(g∘f)(11)=11

    91.

    c

    93.

    A(t)=π(25t+2−−−−√)2A(t)=π(25t+2)2 and A(2)=π(254–√)2=2500πA(2)=π(254)2=2500π square inches

    95.

    A(5)=π(2(5)+1)2=121πA(5)=π(2(5)+1)2=121π square units

    97.

    • ⓐ N(T(t))=23(5t+1.5)2−56(5t+1.5)+1;N(T(t))=23(5t+1.5)2−56(5t+1.5)+1;
    • ⓑ 3.38 hours

    1.5 Section Exercises

    1.

    A horizontal shift results when a constant is added to or subtracted from the input. A vertical shifts results when a constant is added to or subtracted from the output.

    3.

    A horizontal compression results when a constant greater than 1 is multiplied by the input. A vertical compression results when a constant between 0 and 1 is multiplied by the output.

    5.

    For a function f,f, substitute (−x)(−x) for (x)(x) in f(x).f(x). Simplify. If the resulting function is the same as the original function, f(−x)=f(x),f(−x)=f(x), then the function is even. If the resulting function is the opposite of the original function, f(−x)=−f(x),f(−x)=−f(x), then the original function is odd. If the function is not the same or the opposite, then the function is neither odd nor even.

    7.

    g(x)=|x−1|−3g(x)=|x-1|−3

    9.

    g(x)=1(x+4)2+2g(x)=1(x+4)2+2

    11.

    The graph of f(x+43)f(x+43) is a horizontal shift to the left 43 units of the graph of f.f.

    13.

    The graph of f(x−4)f(x-4) is a horizontal shift to the right 4 units of the graph of f.f.

    15.

    The graph of f(x)+8f(x)+8 is a vertical shift up 8 units of the graph of f.f.

    17.

    The graph of f(x)−7f(x)−7 is a vertical shift down 7 units of the graph of f.f.

    19.

    The graph of f(x+4)−1f(x+4)−1 is a horizontal shift to the left 4 units and a vertical shift down 1 unit of the graph of f.f.

    21.

    decreasing on (−∞,−3)(−∞,−3) and increasing on (−3,∞)(−3,∞)

    23.

    decreasing on [0,∞)[0,∞)

    25.

    Graph of k(x).

    27.

    Graph of f(t).

    29.

    Graph of k(x).

    31.

    g(x)=f(x−1),h(x)=f(x)+1g(x)=f(x-1),h(x)=f(x)+1

    33.

    f(x)=|x−3|−2f(x)=|x-3|−2

    35.

    f(x)=x+3−−−−√−1f(x)=x+3−1

    37.

    f(x)=(x−2)2f(x)=(x-2)2

    39.

    f(x)=|x+3|−2f(x)=|x+3|−2

    41.

    f(x)=−x−−√f(x)=−x

    43.

    f(x)=−(x+1)2+2f(x)=−(x+1)2+2

    45.

    f(x)=−x−−−√+1f(x)=−x+1

    47.

    even

    49.

    odd

    51.

    even

    53.

    The graph of gg is a vertical reflection (across the xx-axis) of the graph of f.f.

    55.

    The graph of gg is a vertical stretch by a factor of 4 of the graph of f.f.

    57.

    The graph of gg is a horizontal compression by a factor of 1515 of the graph of f.f.

    59.

    The graph of gg is a horizontal stretch by a factor of 3 of the graph of f.f.

    61.

    The graph of gg is a horizontal reflection across the yy-axis and a vertical stretch by a factor of 3 of the graph of f.f.

    63.

    g(x)=|−4x|g(x)=|−4x|

    65.

    g(x)=13(x+2)2−3g(x)=13(x+2)2−3

    67.

    g(x)=12(x−5)2+1g(x)=12(x-5)2+1

    69.

    The graph of the function f(x)=x2f(x)=x2 is shifted to the left 1 unit, stretched vertically by a factor of 4, and shifted down 5 units.

    Graph of a parabola.71.

    The graph of f(x)=|x|f(x)=|x| is stretched vertically by a factor of 2, shifted horizontally 4 units to the right, reflected across the horizontal axis, and then shifted vertically 3 units up.

    Graph of an absolute function.73.

    The graph of the function f(x)=x3f(x)=x3 is compressed vertically by a factor of 12.12.

    Graph of a cubic function.75.

    The graph of the function is stretched horizontally by a factor of 3 and then shifted vertically downward by 3 units.

    Graph of a cubic function.77.

    The graph of f(x)=x−−√f(x)=x is shifted right 4 units and then reflected across the vertical line x=4.x=4.

    Graph of a square root function.79.

    Graph of a polynomial.

    81.

    Graph of a polynomial.

    1.6 Section Exercises

    1.

    Isolate the absolute value term so that the equation is of the form |A|=B.|A|=B. Form one equation by setting the expression inside the absolute value symbol, A,A, equal to the expression on the other side of the equation, B.B. Form a second equation by setting AA equal to the opposite of the expression on the other side of the equation, −B.−B. Solve each equation for the variable.

    3.

    The graph of the absolute value function does not cross the xx-axis, so the graph is either completely above or completely below the xx-axis.

    5.

    First determine the boundary points by finding the solution(s) of the equation. Use the boundary points to form possible solution intervals. Choose a test value in each interval to determine which values satisfy the inequality.

    7.

    |x+4|=12| x+4 |=12

    9.

    |f(x)−8|<0.03|f(x)−8|<0.03

    11.

    {1,11}{ 1,11 }

    13.

    {−94,134}{ -94,134 }

    15.

    {103,203}{ 103,203 }

    17.

    {115,295}{ 115,295 }

    19.

    {52,72}{ 52,72 }

    21.

    No solution

    23.

    {−57,27}{ −57,27 }

    25.

    (0,−8);(−6,0),(4,0)(0,−8);(−6,0),(4,0)

    27.

    (0,−7);(0,−7); no xx-intercepts

    29.

    (−∞,−8)∪(12,∞)(−∞,−8)∪(12,∞)

    31.

    −43≤x≤4−43≤x≤4

    33.

    (−∞,−83]∪[6,∞)( −∞,−83 ]∪[ 6,∞ )

    35.

    (−∞,−83]∪[16,∞)( −∞,−83 ]∪[ 16,∞ )

    37.

    Graph of an absolute function with points at (-1, 2), (0, 1), (1, 0), (2, 1), and (3, 2).

    39.

    Graph of an absolute function with points at (-2, 3), (-1, 2), (0, 1), (1, 2), and (2, 3).

    41.

    Graph of an absolute function.

    43.

    Graph of an absolute function.

    45.

    Graph of an absolute function.

    47.

    Graph of an absolute function.

    49.

    Graph of an absolute function.

    51.

    Graph of an absolute function.

    53.

    range: [0,20][ 0,20 ]

    Graph of an absolute function.55.

    x-x- intercepts:

    Graph of an absolute function.57.

    (−∞,∞)(−∞,∞)

    59.

    There is no solution for aa that will keep the function from having a yy-intercept. The absolute value function always crosses the yy-intercept when x=0.x=0.

    61.

    |p−0.08|≤0.015| p−0.08 |≤0.015

    63.

    |x−5.0|≤0.01| x−5.0 |≤0.01

    1.7 Section Exercises

    1.

    Each output of a function must have exactly one output for the function to be one-to-one. If any horizontal line crosses the graph of a function more than once, that means that yy-values repeat and the function is not one-to-one. If no horizontal line crosses the graph of the function more than once, then no yy-values repeat and the function is one-to-one.

    3.

    Yes. For example, f(x)=1xf(x)=1x is its own inverse.

    5.

    Given a function y=f(x),y=f(x), solve for xx in terms of y.y. Interchange the xx and y.y. Solve the new equation for y.y. The expression for yy is the inverse, y=f−1(x).y=f−1(x).

    7.

    f−1(x)=x−3f−1(x)=x−3

    9.

    f−1(x)=2−xf−1(x)=2−x

    11.

    f−1(x)=−2xx−1f−1(x)=−2xx−1

    13.

    domain of f(x):[−7,∞);f−1(x)=x−−√−7f(x):[−7,∞);f−1(x)=x−7

    15.

    domain of f(x):[0,∞);f−1(x)=x+5−−−−√f(x):[0,∞);f−1(x)=x+5

    16.

    • ⓐ f(g(x))=xf(g(x))=x and g(f(x))=x.g(f(x))=x.
    • ⓑ This tells us that ff and gg are inverse functions

    17.

    f(g(x))=x,g(f(x))=xf(g(x))=x,g(f(x))=x

    19.

    one-to-one

    21.

    one-to-one

    23.

    not one-to-one

    25.

    33

    27.

    22

    29.

    Graph of a square root function and its inverse.

    31.

    [2,10][ 2,10 ]

    33.

    66

    35.

    −4−4

    37.

    00

    39.

    11

    41.

    xx 1 4 7 12 16
    f−1(x)f−1(x) 3 6 9 13 14

    43.

    f−1(x)=(1+x)1/3f−1(x)=(1+x)1/3

    Graph of a cubic function and its inverse.45.

    f−1(x)=59(x−32).f−1(x)=59(x−32). Given the Fahrenheit temperature, x,x, this formula allows you to calculate the Celsius temperature.

    47.

    t(d)=d50,t(d)=d50, t(180)=18050.t(180)=18050. The time for the car to travel 180 miles is 3.6 hours.

    Review Exercises

    1.

    function

    3.

    not a function

    5.

    f(−3)=−27;f(−3)=−27; f(2)=−2;f(2)=−2; f(−a)=−2a2−3a;f(−a)=−2a2−3a;
    −f(a)=2a2−3a;−f(a)=2a2−3a; f(a+h)=−2a2+3a−4ah+3h−2h2f(a+h)=−2a2+3a−4ah+3h−2h2

    7.

    one-to-one

    9.

    function

    11.

    function

    13.

    Graph of f(x).

    15.

    22

    17.

    x=−1.8x=−1.8 or or x=1.8 or x=1.8

    19.

    −64+80a−16a2−1+a=−16a+64−64+80a−16a2−1+a=−16a+64

    21.

    (−∞,−2)∪(−2,6)∪(6,∞)(−∞,−2)∪(−2,6)∪(6,∞)

    23.

    Graph of f(x).

    25.

    3131

    27.

    increasing (2,∞);(2,∞); decreasing (−∞,2)(−∞,2)

    29.

    increasing (−3,1);(−3,1); constant (−∞,−3)∪(1,∞)(−∞,−3)∪(1,∞)

    31.

    local minimum (−2,−3);(−2,−3); local maximum (1,3)(1,3)

    33.

    Absolute Maximum: 10

    35.

    (f∘g)(x)=17−18x;(g∘f)(x)=−7−18x(f∘g)(x)=17−18x;(g∘f)(x)=−7−18x

    37.

    (f∘g)(x)=1x+2−−−−−√;(f∘g)(x)=1x+2;(g∘f)(x)=1x+2√(g∘f)(x)=1x+2

    39.

    (f∘g)(x)=1+x1+4x,x≠0,x≠−14(f∘g)(x)=1+x1+4x,x≠0,x≠−14

    41.

    (f∘g)(x)=1x√,x>0(f∘g)(x)=1x,x>0

    43.

    sample: g(x)=2x−13x+4;f(x)=x−−√g(x)=2x−13x+4;f(x)=x

    45.

    Graph of f(x)

    47.

    Graph of f(x)

    49.

    Graph of f(x)

    51.

    Graph of f(x)

    53.

    Graph of a half circle.

    55.

    f(x)=|x−3|f(x)=| x−3 |

    57.

    even

    59.

    odd

    61.

    even

    63.

    f(x)=12|x+2|+1f(x)=12| x+2 |+1

    65.

    f(x)=−3|x−3|+3f(x)=−3| x−3 |+3

    67.

    Graph of f(x).

    69.

    x=−22,x=14x=−22,x=14

    71.

    (−53,3)(−53,3)

    73.

    f−1(x)=x−1−−−−√f−1(x)=x-1

    77.

    The function is one-to-one.

    14209d666da3f5e92f51ae26c77a5f27044a81fd78.

    The function is not one-to-one.

    29c44bb73d0d59e7b162343f0579c596c7881d4779.

    55

    Practice Test

    1.

    The relation is a function.

    3.

    −16

    5.

    The graph is a parabola and the graph fails the horizontal line test.

    7.

    2a2−a2a2−a

    9.

    −2(a+b)+1−2(a+b)+1

    11.

    2–√2

    13.

    Graph of f(x).

    15.

    eveneven

    17.

    oddodd

    19.

    x=−7x=−7 and x=10x=10

    21.

    f−1(x)=x+53f−1(x)=x+53

    23.

    (−∞,−1.1) and (1.1,∞)(−∞,−1.1) and (1.1,∞)

    25.

    (1.1,−0.9)(1.1,−0.9)

    27.

    f(2)=2f(2)=2

    29.

    f(x)={|x|ifx≤23ifx>2f(x)={ | x |ifx≤23ifx>2

    31.

    x=2x=2

    33.

    yes

    35.

    f−1(x)=−x−112f−1(x)=−x−112


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