Question 1

Which of the following statements best describes the relationship between a function $$f$$ and its inverse $$f^{-1}$$?

Accepted Answer

The domain of $$f is $$equal to the range of $$f^{-1}$$, and the range of $$f is $$equal to the domain of $$f^{-1}.$$

Question 2

Given $$f(x) = 2x + 3$$, what is $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x-3}{2}$$

Question 3

Which of the following functions does NOT have an inverse that is also a function?

Accepted Answer

$$f(x) = x^2$

Question 4

If $$f(x) = 3x - 4$$ and $$g(x) = \frac{x+4}{3}$$, verify if $$g is $$the inverse of $$f by $$evaluating $$f(g(x)).$$

Accepted Answer

$$f(g(x)) = x$$

Question 5

Which test can be used to determine if a function has an inverse that is also a function?

Accepted Answer

The horizontal line test

Question 6

Suppose the graph of $$f$$ passes the horizontal line test. What does this imply?

Accepted Answer

$$f is $$one-to-one and has an inverse that is a function.

Question 7

Given $$f(x) = \sqrt{x-1}$$, what is the domain of $$f^{-1}(x$$)$$?

Accepted Answer

$$x \geq 0 ($$since the range of $$f is$$ $$[0, \infty)$$, which becomes the domain of $$f^{-1})$$

Question 8

If the point $$(3, 7) is on $$the graph of $$f$$, which point must be on the graph of $$f^{-1}$$?

Accepted Answer

$$(7, 3)$$

Question 9

Which of the following best describes the graphical relationship between $$f$$ and $$f^{-1}$$?

Accepted Answer

The graph of $$f^{-1} is a $$reflection of the graph of $$f$$ about the line $$y = x.$$

Question 10

A medical thermometer converts Celsius to Fahrenheit using $$F(C) = \frac{9}{5}C + 32. $$What is the inverse function, converting Fahrenheit to Celsius?

Accepted Answer

$$C(F) = \frac{5}{9}(F - 32)$$

Question 11

If $$f(x) = \frac{1}{x}$$, what is $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{1}{x}$$

Question 12

A car rental company charges $$f(x) = 40x + 25$$ dollars for $$x$$ days of rental. If you paid $$185$$, for how many days did you rent the car?

Accepted Answer

4 days

Question 13

Which of the following is a necessary condition for a function $$f to $$have an inverse that is also a function?

Accepted Answer

$$f$$ must be one-to-one.

Question 14

Given $$f(x) = 5 - 2x$$, find $$f^{-1}(x$$)$$.

Accepted Answer

$$f^{-1}(x$$) = \frac{5 - x}{-2}$$

Question 15

If $$f(x) = x^3$ + 1$$, what is $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \sqrt[3]{x - 1}$$

Inverse Functions: Definitions, Properties, and Graphical Analysis

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