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 $$f(x) = 5x - 7$$ and $$g(x) = \frac{x + 7}{5}$$, 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

Horizontal Line Test

Question 6

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

Accepted Answer

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

Question 7

Given $$f(x) = 3x - 4$$, find $$f^{-1}(2$$)$$.

Accepted Answer

$$\frac{2 + 4}{3} = 2$$

Question 8

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

Accepted Answer

$$(b, a)$$

Question 9

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

Accepted Answer

They are reflections of each other about the line $$y = x.$$

Question 10

A real-world scenario: The temperature $$T in $$Celsius is related to the temperature $$F in $$Fahrenheit by $$F = \frac{9}{5}C + 32. $$What is the inverse function to convert Fahrenheit to Celsius?

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

If $$f(x) = x^2$ for x$$ \geq 0$$, what is f$$^{-1}$$(x)$$?

Accepted Answer

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

Question 13

A hospital uses the function $$f(x) = 0.8x + 50 to $$convert a patient's heart rate $$x (in $$beats per minute) to a risk score. What is the inverse function to find the heart rate from a risk score?

Accepted Answer

$$f^{-1}(x$$) = \frac{x - 50}{0.8}$$

Question 14

Which of the following is a necessary step in finding the inverse of a function $$f(x)$$?

Accepted Answer

Replace $$f(x)$$ with $$y$$, interchange $$x$$ and $$y$$, and solve for $$y.$$

Inverse Functions: Definitions, Properties, and Graphical Analysis

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