Question 1

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

Accepted Answer

A function where each $$x$$ value corresponds to only one $$y$$ value, and each $$y$$ value corresponds to only one $$x$$ value.

Question 2

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

Accepted Answer

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

Question 3

Suppose the concentration of a drug in a patient’s blood is modeled by $$C = f(t)$$, where $$t is $$time in hours. If you want to know how long it takes for the concentration to reach a certain value $$C_0$$, which function do you use?

Accepted Answer

Use the inverse function $$f^{-1}$$($$C_0$$)$$ to solve for t$.

Question 4

Which of the following functions is NOT one-to-one on its natural domain?

Accepted Answer

$$f(x) = x^2$

Question 5

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

Accepted Answer

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

Question 6

Which test can be used to determine if a function is one-to-one?

Accepted Answer

Horizontal Line Test

Question 7

Given $$f(x) = \sqrt{x}$$, what is the domain of $$f^{-1}(x$$)$$?

Accepted Answer

$$x \geq 0$$

Question 8

If $$f(x) = 8x - 2$$ and $$g(x) = \frac{1}{8}x + \frac{1}{4}$$, what is $$f(g(x))$$?

Accepted Answer

$$x$$

Question 9

Find the inverse of $$f(x) = \frac{2x-1}{3x+2}.$$

Accepted Answer

Question 10

Given $$f(x) = (x-3)^2$ defined on x$$ \geq 3$$, what is f$$^{-1}$$(x)$$?

Accepted Answer

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

Question 11

If $$f(x)$$ and $$g(x)$$ are inverses, which of the following must be true for all $$x in $$the domain of $$f$$?

Accepted Answer

Both $$f(g(x)) = x$$ and $$g(f(x)) = x$$

Question 12

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

Accepted Answer

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

Question 13

Given $$f(x) = \frac{2}{x+5}$$, which of the following is $$f^{-1}(x$$)$$?

Accepted Answer

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

Question 14

If the domain of $$f is$$ $$[0, \infty)$$ and the range is $$[2, \infty)$$, what is the domain of $$f^{-1}$$?

Accepted Answer

$$[2, \infty)$$

Question 15

A real-world scenario: The temperature $$T in $$Celsius is related to the temperature $$F in $$Fahrenheit 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 16

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

Accepted Answer

The function must be one-to-one.

Question 17

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

Accepted Answer

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

Question 18

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

Accepted Answer

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

Question 19

A function $$f is $$invertible if and only if:

Accepted Answer

It passes the horizontal line test.

Inverse Functions and One-to-One Functions (Section 4.1 Study Notes)

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