1.7E: Exercises
- Page ID
- 31074
This page is a draft and is under active development.
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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
- Answer:
- not one-to-one
Exercise 1.7.24
For the following exercises, use the graph of \(f\) shown in Figure 1.7.11.
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.
Figure 1.7.12: Graph of a square root function.
Exercise 1.7.29
Sketch the graph of \(f^{-1}\).
- Answer:
-
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:
-
\(f^{-1}(x)=(1+x)^{1/3}\)
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.