Q: A function $f$ is one-to-one if and only if:

Different inputs have different outputs: if $a \neq b$, then $f(a) \neq f(b)$.

Question 1

Which of the following statements correctly describes the relationship between a function and its inverse?

Accepted Answer

The graph of a function and its inverse are always reflections of each other across the line $y = x$.

Question 2

Given the function $f(x) = 2x - 3$, which of the following is the correct formula for its inverse $f^{-1}(x)$?

Accepted Answer

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

Question 3

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

Accepted Answer

$f(x) = x^2$

Question 4

A function $f$ is one-to-one if and only if:

Accepted Answer

Different inputs have different outputs: if $a 
eq b$, then $f(a) 
eq f(b)$.

Question 5

If $f(x) = 5x + 8$, what is $f^{-1}(x)$?

Accepted Answer

$f^{-1}(x) = \frac{x - 8}{5}$

Question 6

Which test can be used to determine if a function is one-to-one by looking at its graph?

Accepted Answer

Horizontal-line test

Question 7

Suppose $f(x) = x^2$ and its domain is restricted to $x \geq 0$. What is the inverse function $f^{-1}(x)$?

Accepted Answer

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

Question 8

Given $f(x) = 2x + 5$ and $g(x) = x - 3$, what is $(f \circ g)(x)$?

Accepted Answer

$(f \circ g)(x) = 2(x - 3) + 5$

Question 9

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

Accepted Answer

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

Question 10

Which of the following is true about the notation $f^{-1}(x)$?

Accepted Answer

$f^{-1}(x)$ is the inverse function of $f(x)$, not the reciprocal.

Question 11

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

Accepted Answer

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

Question 12

If $f(x) = x^2 - 2x$, what is the equation for its inverse relation?

Accepted Answer

$x = y^2 - 2y$

Question 13

Which of the following statements is true about the domain and range of a one-to-one function and its inverse?

Accepted Answer

The domain of $f$ is the range of $f^{-1}$, and the range of $f$ is the domain of $f^{-1}$.

Question 14

A chemical process is modeled by $f(x) = 4x + 10$. If the output is 34, what was the input?

Accepted Answer

6

Question 15

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

Accepted Answer

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

Question 16

If $f(x) = x^2$ and $f^{-1}(x) = \sqrt{x}$, what domain restriction is necessary for $f^{-1}(x)$ to be a function?

Accepted Answer

$x \geq 0$

Question 17

Which of the following compositions always returns the input $x$ for a one-to-one function $f$ and its inverse $f^{-1}$?

Accepted Answer

$(f \circ f^{-1})(x)$ and $(f^{-1} \circ f)(x)$

Question 18

A medication dosage is calculated by $f(x) = 0.5x + 2$, where $x$ is the patient's weight in kg. If a patient receives a dosage of 7 mg, what is their weight?

Accepted Answer

10 kg

Question 19

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

Accepted Answer

A horizontal line can intersect the graph more than once.

Inverse Functions and One-to-One Functions: Study Notes for College Algebra

Study Guide - Practice Questions

Study Guide - Flashcards