Question 1

Which of the following best describes the relationship between a function and its inverse?

Accepted Answer

They undo each other's operations, so $$f(g(x)) = x$$ and $$g(f(x)) = x.$$

Question 2

Given the function $$f(x) = 2x + 5$$, what is the equation of its inverse $$f^{-1}(x$$)$$?

Accepted Answer

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

Question 3

Which of the following functions is NOT one-to-one?

Accepted Answer

$$f(x) = 25 - x^2$

Question 4

Refer to the first image. Does the function shown pass the horizontal line test?

Accepted Answer

No, it does not pass the horizontal line test.

Question 5

If $$f(x) = x^3$ - 1$$ and $$g(x) = \sqrt[3]{x+1}$$, what is $$(f \circ g)(x)$$?

Accepted Answer

$$(f \circ g)(x) = x$$

Question 6

Which of the following statements about the inverse of a function is TRUE?

Accepted Answer

The range of $$f is $$the domain of $$f^{-1}.$$

Question 7

Given the set $$g = \{(3, 1), (0, 2), (2, 3), (4, 0)\}$$, what is $$g^{-1}$$?

Accepted Answer

$$g^{-1}$$ = \{(1, 3), (2, 0), (3, 2), (0, 4)\}$$

Question 8

Refer to the second image. Does the function $$f(x) = x^3$ - 1$$ pass the horizontal line test?

Accepted Answer

Yes, it passes the horizontal line test.

Question 9

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 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

A real-world scenario: A temperature conversion function is given by $$f(C) = 1.8C + 32$$, where $$C is $$degrees Celsius and $$f(C) is $$degrees Fahrenheit. What is the inverse function, which converts Fahrenheit back to Celsius?

Accepted Answer

$$f^{-1}(F$$) = \frac{F-32}{1.8}$$

Question 13

Suppose $$f(x) = (x-2)^3$. What is f$$^{-1}$$(x)$$?

Accepted Answer

$$f^{-1}(x$$) = \sqrt[3]{x} + 2$$

Question 14

Refer to the last image. What is $$f^{-1}(0$$)$$ for the function shown?

Accepted Answer

$$-2$$

Question 15

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 16

If $$f(x) is a $$one-to-one function and $$f(3) = 7$$, what is $$f^{-1}(7$$)$$?

Accepted Answer

$$3$$

Question 17

A function $$f is $$defined as $$f(x) = x^2$ + 2$$ for $$x \geq 0. $$What is the inverse $$f^{-1}(x$$)$$?

Accepted Answer

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

Question 18

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

Accepted Answer

Converting miles to kilometers and then back to miles.

Question 19

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

Accepted Answer

Every horizontal line intersects the graph of $$f at $$most once.

Inverse Functions and One-to-One Functions: Precalculus Study Notes

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