Question 1

Which of the following statements best describes a one-to-one function?

Accepted Answer

A function where any two different inputs in the domain correspond to two different outputs in the range.

Question 2

Refer to the first image. What does the horizontal line test determine about the function $$y = f(x)$$ shown?

Accepted Answer

The function is not one-to-one because there exists a horizontal line that intersects the graph more than once.

Question 3

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

Accepted Answer

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

Question 4

If $$f(x) is a $$one-to-one function, what is true about the graph of $$f^{-1}(x$$)$$?

Accepted Answer

It is symmetric with respect to the line $$y = x.$$

Question 5

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

Accepted Answer

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

Question 6

Why must the domain of $$f(x) = x^2$ be restricted to x$$ \geq 0$$ to have an inverse?

Accepted Answer

Because otherwise the function is not one-to-one and fails the horizontal line test.

Question 7

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

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

Switch $$x$$ and $$y in $$the equation.

Question 10

Suppose $$f(x) is a $$one-to-one function with domain $$D$$ and range $$R. $$What are the domain and range of $$f^{-1}(x$$)$$?

Accepted Answer

Domain $$R$$, Range $$D$$

Question 11

If $$f(x) = 4x - 7$$, what is $$f^{-1}(x$$)$$?

Accepted Answer

$$f^{-1}(x$$) = \frac{x+7}{4}$$

Question 12

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 13

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

Accepted Answer

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

Question 14

If $$f(x) is $$increasing on an interval $$I$$, what can be said about $$f on$$ $$I$$?

Accepted Answer

$$f is $$one-to-one on $$I.$$

Question 15

Which of the following is NOT a property of inverse functions?

Accepted Answer

$$f^{-1}(x$$)$$ is always defined for all real numbers.

Question 16

A hospital uses a formula $$C = 1.8F + 32 to $$convert a patient's temperature from Fahrenheit $$F to $$Celsius $$C. $$What is the inverse formula to convert Celsius to Fahrenheit?

Accepted Answer

$$F = \frac{C - 32}{1.8}$$

Inverse Functions: One-to-One Functions and Their Inverses

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