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Mathematics LibreTexts

1.7E: Exercises

  • Page ID
    17377
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    Section Exercises

    Verbal

    Exercise 1.7.1

    Describe why the horizontal line test is an effective way to determine whether a function is one-to-one?

    Answer:
    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 \(y\)-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 \(y\)-values repeat and the function is one-to-one.

    Exercise 1.7.2

    Why do we restrict the domain of the function \(f(x)=x^2\) to find the function’s inverse?

    Exercise 1.7.3

    Can a function be its own inverse? Explain.

    Answer:
    Yes. For example, \(f(x)=\frac{1}{x}\) is its own inverse.

    Exercise 1.7.4

    Are one-to-one functions either always increasing or always decreasing? Why or why not?

    Exercise 1.7.5

    How do you find the inverse of a function algebraically?

    Answer:
    Given a function \(y=f(x)\), solve for \(x\) in terms of \(y\). Interchange the \(x\) and \(y\). Solve the new equation for \(y\). The expression for \(y\) is the inverse, \(y=f^{-1}(x)\).

    Algebraic

    Exercise 1.7.6

    Show that the function \(f(x)=a−x\) is its own inverse for all real numbers \(a\).

    For the following exercises, find \(f^{-1}(x)\) for each function.

    Exercise 1.7.7

    \(f(x)=x+3\)

    Answer:
    \(f^{-1}(x)=x−3\)

    Exercise 1.7.8

    \(f(x)=x+5\)

    Exercise 1.7.9

    \(f(x)=2−x\)

    Answer:
    \(f^{-1}(x)=2−x\)

    Exercise 1.7.10:

    \(f(x)=3−x\)

    Exercise 1.7.11

    \(f(x)=\frac{x}{x+2}\)

    Answer:
    \(f^{-1}(x)=\frac{−2x}{x−1}\)

    Exercise 1.7.12

    \(f(x)=\frac{2x+3}{5x+4}\)

    For the following exercises, find a domain on which each function \(f\) is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of \(f\) restricted to that domain.

    Exercise 1.7.13

    \(f(x)=(x+7)^2\)

    Answer:
    domain of \(f(x)\): \(\left[−7,\infty\right)\); \(f^{-1}(x)=\sqrt{x}−7\)

    Exercise 1.7.14

    \(f(x)=(x−6)^2\)

    Exercise 1.7.15

    \(f(x)=x^2−5\)

    Answer:
    domain of \(f(x)\): \(\left[0,\infty\right)\); \(f^{-1}(x)=\sqrt{x+5}\)

    Exercise 1.7.16

    Given \(f(x)=\frac{x}{2+x}\) and \(g(x)=\frac{2x}{1-x}:\)

    a. Find \(f(g(x))\) and \(g(f(x))\).

    b. What does the answer tell us about the relationship between \(f(x)\) and \(g(x)?\)

    Answer:
    a. \(f(g(x))=x\) and \(g(f(x))=x\).

    b. This tells us that \(f\) and \(g\) are inverse functions

    For the following exercises, use function composition to verify that \(f(x)\) and \(g(x)\) are inverse functions.

    Exercise 1.7.17

    \(f(x)=\sqrt[3]{x-1}\) and \(g(x)=x^3+1\)

    Answer:
    \(f(g(x))=x\), \(g(f(x))=x\)

    Exercise 1.7.18

    \(f(x)=−3x+5\) and \(g(x)=\frac{x-5}{-3}\)

    Graphical

    For the following exercises, use a graphing utility to determine whether each function is one-to-one.

    Exercise 1.7.19

    \(f(x)=\sqrt{x}\)

    Answer:
    one-to-one

    Exercise 1.7.20

    \(f(x)=\sqrt[3]{3x+1}\)

    Exercise 1.7.21

    \(f(x)=−5x+1\)

    Answer:
    one-to-one

    Exercise 1.7.22

    \(f(x)=x^3−27\)

    For the following exercises, determine whether the graph represents a one-to-one function.

    Exercise 1.7.23

    Graph of a parabola.

    Answer:
    not one-to-one

    Exercise 1.7.24

    Graph of a step-function.

    For the following exercises, use the graph of \(f\) shown in Figure 1.7.11.

    Graph of a line.

    Figure 1.7.11: Graph of a line

    Exercise 1.7.25

    Find \(f(0)\).

    Answer:
    3

    Exercise 1.7.26

    Solve \(f(x)=0\).

    Exercise 1.7.27

    Find \(f^{-1}(0)\).

    Answer:
    2

    Exercise 1.7.28

    Solve \(f^{-1}(x)=0\).

    For the following exercises, use the graph of the one-to-one function shown in Figure 1.7.12.

    Graph of a square root function.
    Figure 1.7.12: Graph of a square root function.

    Exercise 1.7.29

    Sketch the graph of \(f^{-1}\).

    Answer:

    Graph of a square root function and its inverse.

    Exercise 1.7.30

    Find \(f(6)\) and \(f^{-1}(2)\).

    Exercise 1.7.31

    If the complete graph of \(f\) is shown, find the domain of \(f\).

    Answer:
    \([2,10]\)

    Exercise 1.7.32

    If the complete graph of \(f\) is shown, find the range of \(f\)

    Numeric

    For the following exercises, evaluate or solve, assuming that the function \(f\) is one-to-one.

    Exercise 1.7.33

    If \(f(6)=7\), find \(f^{-1}(7)\).

    Answer:
    6

    Exercise 1.7.34

    If \(f(3)=2\), find \(f^{-1}(2)\).

    Exercise 1.7.35

    If \(f^{-1}(−4)=−8\), find \(f(−8)\).

    Answer:
    -4

    Exercise 1.7.36

    If \(f^{-1}(−2)\)=−1, find \(f(−1)\).

    For the following exercises, use the values listed in Table 1.7.6 to evaluate or solve.

    \(x\) \(f(x)\)
    0 8
    1 0
    2 7
    3 4
    4 2
    5 6
    6 5
    7 3
    8 9
    9 1

    Table 1.7.6

    Exercise 1.7.37

    Find \(f(1)\).

    Answer:
    0

    Exercise 1.7.38

    Solve \(f(x)=3\).

    Exercise 1.7.39

    Find \(f^{-1}(0)\).

    Answer:
    1

    Exercise 1.7.40

    Solve \(f^{-1}(x)=7\).

    Exercise 1.7.41

    Use the tabular representation of \(f\) in Table 1.7.7 to create a table for f^{-1}(x).

    \(x\)

    3 6 9 13 14

    \(f(x)\)

    1 4 7 12 16

    Table 1.7.7

    Answer:

    \(x\)

    1 4 7 12 16

    \(f^{-1}(x)\)

    3 6 9 13 14

    Technology

    For the following exercises, find the inverse function. Then, graph the function and its inverse.

    Exercise 1.7.42

    \(f(x)=\dfrac{3}{x-2}\)

    Exercise 1.7.43

    \(f(x)=x^3−1\)

    Answer:

    \(f^{-1}(x)=(1+x)^{1/3}\)

    Graph of a cubic function and its inverse.

    Exercise 1.7.44

    Find the inverse function of \(f(x)=\frac{1}{x-1}\). Use a graphing utility to find its domain and range. Write the domain and range in interval notation.

    Real-World Applications

    Exercise 1.7.45

    To convert from \(x\) degrees Celsius to \(y\) degrees Fahrenheit, we use the formula \(f(x)=\frac{9}{5}x+32\). Find the inverse function, if it exists, and explain its meaning.

    Answer:
    \(f^{-1}(x)=\frac{5}{9}(x−32)\). Given the Fahrenheit temperature, \(x\), this formula allows you to calculate the Celsius temperature.

    Exercise 1.7.46

    The circumference \(C\) of a circle is a function of its radius given by \(C(r)=2{\pi}r\). Express the radius of a circle as a function of its circumference. Call this function \(r(C)\). Find \(r(36\pi)\) and interpret its meaning.

    Exercise 1.7.47

    A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, \(t\), in hours given by \(d(t)=50t\). Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function \(t(d)\). Find \(t(180)\) and interpret its meaning.

    Answer:
    \(t(d)=\frac{d}{50}\), \(t(180)=\frac{180}{50}\). The time for the car to travel 180 miles is 3.6 hours.