Precalculus Inverse Functions Worksheet (PDF)

Precalculus math worksheet: Inverse Functions – Maths PDF

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This inverse functions worksheet for Precalculus covers inverse functions (Common Core F-BF.B.4a, F-BF.B.4b, F-BF.B.4c, F-BF.B.4d), with a free PDF to print.

These 20 problems practice inverse functions (Common Core F-BF.B.4a through F-BF.B.4d). You will test for one-to-one functions, find inverses algebraically, verify with compositions and restrict domains.

Precalculus · Common Core State Standards: F-BF.B.4a, F-BF.B.4b, F-BF.B.4c, F-BF.B.4d

Inverse functions worksheet: exercises 1 to 10

Exercise 1 – One-to-one table

Table of f.

Is $f$ one-to-one? Explain.

Exercise 2 – Inverse of a line

Find $f^{-1}(x)$ for $f(x)=3x-6$.

Exercise 3 – Inverse of a quotient by a constant

Find $f^{-1}(x)$ for $f(x)=\frac{x+4}{5}$.

Exercise 4 – Inverse of a cubic

Find $f^{-1}(x)$ for $f(x)=x^{3}+1$.

Exercise 5 – Inverse of a rational function

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

Exercise 6 – Inverse of a root

Find $f^{-1}(x)$ for $f(x)=\sqrt{x-2}$ and state its domain.

Exercise 7 – Inverse of a restricted parabola

Find the inverse of $f(x)=(x-3)^{2}$ for $x\ge 3$.

Exercise 8 – Verify inverses

Show that $f(x)=4x+7$ and $g(x)=\frac{x-7}{4}$ are inverses.

Exercise 9 – Read a table of an inverse

Table of f.

Use the table to find $f^{-1}(11)$, $f^{-1}(5)$ and $f(f^{-1}(14))$.

Exercise 10 – Equation from a graph

Graph of f(x) = 2x + 1 and its mirror image.

The graph shows $f(x)=2x+1$ (blue) and a second line (orange) that mirrors it over $y=x$. Write the equation of the orange line.

Exercises 11 and up

Exercise 11 – Domain and range of an inverse

For $f(x)=\sqrt{x+1}$, state the domain and range of $f$ and of $f^{-1}$, and find $f^{-1}(x)$.

Exercise 12 – Restrict the domain

Graph of y = x^2.

The function $f(x)=x^{2}-4$ is not one-to-one. Restrict the domain to $x\ge 0$ and find the inverse.

Exercise 13 – Temperature conversion

The formula $F=1.8C+32$ converts Celsius to Fahrenheit. Solve for $C$, then convert $95^{\circ}\text{F}$ to Celsius.

Exercise 14 – Rational inverse and check

Find $f^{-1}(x)$ for $f(x)=\frac{x+2}{x-3}$, then check with $x=5$.

Exercise 15 – Complete the square first

Find the inverse of $f(x)=x^{2}+6x$ for $x\ge -3$.

Exercise 16 – Another rational inverse

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

Exercise 17 – Taxi fare inverse

Fare for 10 miles.

An Austin taxi charges $C(m)=4+2.5m$ dollars for $m$ miles. Find $C^{-1}$, explain what it tells you, and find how far Liam rides for $\$29$.

Exercise 18 – Car value

Value of Mei’s car.

The value of Mei’s car is $V(t)=24{,}000-1{,}800t$ dollars after $t$ years. Find the inverse, and use it to find when the value is $\$15{,}000$ and when it is $\$12{,}000$.

Exercise 19 – Find f from its inverse

A linear function $f$ has inverse $f^{-1}(x)=\frac{x-3}{2}$. Find $f(x)$ and check that $f(1)=5$.

Exercise 20 – Where f meets its inverse

Graph of f and y = x.

Let $f(x)=\frac{x}{2}+3$. The point $(2,4)$ is on $f$. Name the matching point on $f^{-1}$, then find the point where $f$ meets the line $y=x$ (this point is also on $f^{-1}$).

Key facts behind this inverse functions worksheet

Key facts

f^-1(f(x)) = x and f(f^-1(x)) = x.
Find an inverse: write y = f(x), swap x and y, solve for y.
Domain of f becomes the range of f^-1, and the range of f becomes the domain of f^-1.
The graph of f^-1 is the reflection of the graph of f over y = x.
Restrict the domain of a function that is not one-to-one, such as x^2 with x >= 0.

How to use this inverse functions worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the inverse functions worksheet as a PDF with the button below and keep it with your notes.

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