Question 1

Which of the following statements best describes the condition for a function $$f(x) to be $$invertible?

Accepted Answer

It must be one-to-one, meaning it passes the horizontal line test.

Question 2

Given the function $$f(x) = 3x - 4$$, find the inverse function $$f^{-1}(x$$)$$.

Accepted Answer

$$f^{-1}(x$$) = \frac{x+4}{3}$$

Question 3

Suppose $$f is an $$invertible function and $$f^{-1}(10) = 4$. What is f(4$$)$$?

Accepted Answer

$$f(4) = 10$$

Question 4

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

Accepted Answer

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

Question 5

A function $$f is $$defined on the domain $$[-2,4]$$ and its range is $$[-2,3]. $$What is the domain and range of $$f^{-1}$$?

Accepted Answer

Domain: $$[-2,3]$$, Range: $$[-2,4]$$

Question 6

Given the table below for an invertible function $$f$$:

$$x$$: 1, 2, 3, 4, 5
$$f(x)$$: 9, 7, 5, 3, 1

What is $$f^{-1}(3$$)$$?

Accepted Answer

$$f^{-1}(3) = 4$

Question 7

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

Accepted Answer

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

Question 8

Which of the following graphs represents a function that is NOT invertible?

Accepted Answer

A graph that fails the horizontal line test.

Question 9

Suppose $$f$$ and $$g$$ are inverse functions. Which of the following is always true?

Accepted Answer

$$f(g(x)) = x$$ for all $$x in $$the domain of $$g$$

Question 10

Given $$f(x) = \sqrt{x+5}$$ and $$g(x) = x^2$ - 5$$, verify that $$f$$ and $$g$$ are inverse functions.

Accepted Answer

$$f(g(x)) = \sqrt{(x^2$ - 5) + 5} = \sqrt{x^2$} = x$$ and $$g(f(x)) = (\sqrt{x+5})^2$ - 5 = x + 5 - 5 = x$$

Question 11

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

Accepted Answer

$$f(3) = 7$$

Question 12

Given $$f(x) = 2x^3$ - 1$$, find $$f^{-1}(x$$)$$.

Accepted Answer

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

Question 13

A hospital uses a function $$f to $$convert Celsius temperatures to Fahrenheit: $$f(x) = \frac{9}{5}x + 32. $$What is the inverse function $$f^{-1}(x$$)$$ to convert Fahrenheit back to Celsius?

Accepted Answer

$$f^{-1}(x$$) = \frac{5}{9}(x - 32)$$

Question 14

If $$f(x) is $$invertible and $$f^{-1}(a) = b$, which of the following must be true?

Accepted Answer

$$f(b) = a$$

Question 15

A function $$f is $$defined by the table below:

$$x$$: 1, 2, 3, 4, 5
$$f(x)$$: 6, 8, 4, 3, 9

What is $$f^{-1}(4$$)$$?

Accepted Answer

$$f^{-1}(4) = 3$

Question 16

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

Accepted Answer

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

Question 17

Given $$f(x) = \sqrt{x+4} - 4$$, what is $$f^{-1}(x$$)$$?

Accepted Answer

Question 18

A chemical process uses the function $$f(x) = 2x + 5 to $$convert the amount of reactant $$x to $$the amount of product $$y. If $$the product amount is measured as $$y = 17$$, what was the original amount of reactant?

Accepted Answer

$$x = 6$$

Question 19

Which of the following is true about the graph of an invertible function and its inverse?

Accepted Answer

The graph of the inverse is a reflection of the original function across the line $$y = x.$$

Inverse Functions and Their Properties in Precalculus

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