Question 1

Which of the following is the correct definition of the inverse of a function $$f$$?

Accepted Answer

$$f(f^{-1}(x)) = x$ for every x$ in the domain of f$$^{-1}$$ and f$$^{-1}$$(f(x)) = x$$ for every $$x in $$the domain of $$f$$

Question 2

Given $$f(x) = 2x + 7$$, what is $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x-7}{2}$$

Question 3

Which of the following steps is NOT part of the process for finding the inverse of a function $$f(x)$$?

Accepted Answer

Differentiate $$f(x)$$ with respect to $$x$$

Question 4

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

Accepted Answer

$$f^{-1}(x$$) = \sqrt{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 $$f(x)$$ passes the horizontal line test. What does this imply?

Accepted Answer

$$f(x) is $$one-to-one and has an inverse that is a function

Question 7

If $$f(x) = 3x - 4$$ and $$g(x) = \frac{x+4}{3}$$, are $$f$$ and $$g$$ inverses of each other?

Accepted Answer

Yes, because $$f(g(x)) = x$$ and $$g(f(x)) = x$$

Question 8

Which of the following statements is TRUE about the graphs of $$f$$ and $$f^{-1}$$?

Accepted Answer

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

Question 9

A function $$f is $$defined as $$f(x) = \frac{1}{x}. $$What is the domain of $$f^{-1}$$?

Accepted Answer

All real numbers except $$0$$

Question 10

Given the graph of $$y = |x|$$, does this function have an inverse that is also a function?

Accepted Answer

No, because it does not pass the horizontal line test

Question 11

If $$f(x) = 5 - 2x$$, what is $$f^{-1}(x$$)$$?

Accepted Answer

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

Question 12

Which of the following real-world scenarios best illustrates the concept of inverse functions?

Accepted Answer

Converting temperatures from Celsius to Fahrenheit and back

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

The function must be one-to-one

Question 15

Suppose $$f(x) = 4x - 1. $$What is $$f^{-1}(2$$)$$?

Accepted Answer

$$f^{-1}(2$$) = \frac{2+1}{4} = \frac{3}{4}$$

Question 16

If $$f(x) = \frac{x-3}{2}$$, what is $$f^{-1}(x$$)$$?

Accepted Answer

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

Question 17

A function $$f$$ has domain $$[0, \infty)$$ and is defined by $$f(x) = x^2$. Which of the following statements is true?

Accepted Answer

$$f$$ has an inverse function because it is one-to-one on $$[0, \infty)$$

Inverse Functions: Definitions, Properties, and Graphical Analysis

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