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 $f^{-1}(x)$?

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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