$$f(g(x)) = x$$ \text{ and } $$g(f(x)) = x$$

Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

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

Question 3

If $$f(x) = 3x + 4$$, what is the domain of $$f^{-1}(x$$)$$?

Accepted Answer

All real numbers

Question 4

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 pass the horizontal line test.

Question 5

Suppose $$f(x) = 4x - 7$$ and $$g(x) = \frac{x+7}{4}. $$Which of the following statements is true?

Accepted Answer

$$f$$ and $$g$$ are inverses of each other.

Question 6

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

Accepted Answer

Question 7

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

Accepted Answer

All real numbers

Question 8

A function $$f is $$defined by the ordered pairs $$\{(-2, -2), (-1, 0), (1, 2)\}. $$What are the ordered pairs for $$f^{-1}$$?

Accepted Answer

$$\{(-2, -2), (0, -1), (2, 1)\}$$

Question 9

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

Accepted Answer

Converting temperatures from Celsius to Fahrenheit and back.

Question 10

Given $$f(x) = 2x - 3x + 1$$, what is the correct process to find $$f^{-1}(x$$)$$?

Accepted Answer

Replace $$f(x)$$ with $$y$$, interchange $$x$$ and $$y$$, solve for $$y$$, and replace $$y$$ with $$f^{-1}(x$$)$$.

Question 11

If $$f(x) is a $$one-to-one function, which of the following must be true?

Accepted Answer

Each $$y-$$value in the range of $$f$$ corresponds to exactly one $$x-$$value in the domain.

Question 12

Which of the following verifies that $$f$$ and $$g$$ are inverses of each other?

Accepted Answer

$$f(g(x)) = x$$ 	ext{ and } $$g(f(x)) = x$$

Question 13

A hospital uses a function $$f to $$convert a patient's weight from kilograms to pounds. What is the inverse function $$f^{-1}$$ used for?

Accepted Answer

Converting a patient's weight from pounds to kilograms.

Question 14

If $$f(x) = (x-3)^2$ for x$$ \leq 3$$, what is the domain of f$$^{-1}$$(x)$$?

Accepted Answer

$$x \geq 0$$

Question 15

Which of the following is NOT a step in finding the inverse of a function algebraically?

Accepted Answer

Take the derivative of the function.

Question 16

A function $$f$$ has domain $$[0, \infty)$$ and range $$[2, \infty). $$What are the domain and range of $$f^{-1}$$?

Accepted Answer

Domain: $$[2, \infty)$$, Range: $$[0, \infty)$$

Question 17

Which of the following graphs represents a function that does NOT have an inverse function?

Accepted Answer

A graph that fails the horizontal line test.

Question 18

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

Accepted Answer

$$4$$

Question 19

Which of the following best describes the process of verifying that two functions are inverses?

Accepted Answer

Show that $$f(g(x)) = x$$ and $$g(f(x)) = x$$ for all $$x in $$the appropriate domains.

Inverse Functions: Definitions, Properties, and Graphs

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