Q: Which of the following statements best describes when a function $$f(x)$$ has an inverse?

If $$f(x) is $$one-to-one, meaning $$f(x_1$) \ne f(x_2$)$$ whenever $$x_1$$ \ne $$x_2.$$

Question 1

Which of the following statements best describes when a function $$f(x)$$ has an inverse?

Accepted Answer

If $$f(x) is $$one-to-one, meaning $$f(x_1$) 
e f(x_2$)$$ whenever $$x_1$$ 
e $$x_2.$$

Question 2

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

Accepted Answer

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

Question 3

If a particle's velocity is given by $$v(t) = 3t + 2$$, at what time $$t is $$the velocity $$17 m/s$$?

Accepted Answer

$$t = 5$$

Question 4

What is the domain of the inverse cosine function $$\cos$$^{-1}$$(x)$$?

Accepted Answer

$$[-1, 1]$$

Question 5

What is $$\cos$$^{-1}$$\left(\frac{1}{2}\right)$$?

Accepted Answer

$$\frac{\pi}{3}$$

Question 6

What is $$\cos(\sin$$^{-1}$$(1))$$?

Accepted Answer

$$0$$

Question 7

Suppose $$f(x) = \sqrt[4]{x} - 2. $$What is the domain of $$f(x)$$?

Accepted Answer

$$[0, \infty)$$

Question 8

Suppose $$f(x) = \sqrt[4]{x} - 2. $$What is the range of $$f(x)$$?

Accepted Answer

$$[-2, \infty)$$

Question 9

Given $$f(x) = \sqrt[4]{x} - 2$$, what is the domain of its inverse function?

Accepted Answer

$$[-2, \infty)$$

Question 10

Given $$f(x) = \sqrt[4]{x} - 2$$, what is the range of its inverse function?

Accepted Answer

$$[0, \infty)$$

Question 11

Which of the following is NOT a property of an inverse function $$f^{-1}(y$$)$$ for a one-to-one function f(x$$)$$?

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

$$f(x) = x^2$

Question 14

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

Accepted Answer

$$3$$

Question 15

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

Accepted Answer

Replace $$f(x)$$ with $$y$$, solve for $$x in $$terms of $$y$$, then swap $$x$$ and $$y.$$

Question 16

If $$f(x) = 2x - 5$$ and $$g(x) = f$$^{-1}$$(x)$$, what is $$g(9)$$?

Accepted Answer

$$7$$

Inverse Functions and One-to-One Functions: Definitions, Properties, and Examples

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