How to Find Inverse of a Function? (+FREE Worksheet!)

A function inverse reverses the action of the original function: if f takes an input x and produces an output y, then the inverse function f⁻¹ takes y as input and returns x. Understanding function inverses is a key skill in Algebra 1 and beyond, and finding an inverse is as straightforward as swapping x and y in the equation and solving for the new y.

What Is a Function Inverse?

The inverse of a function f is written f⁻¹ (read “f inverse”). It satisfies the conditions:

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• f⁻¹\(\color{blue}{(f(x)) = x}\) for every x in the domain of f
• \(\color{blue}{f(f<\text{ sup }>-1(x)) = x}\) for every x in the domain of f⁻¹

Graphically, the inverse of f is the reflection of f over the line \(\color{blue}{y = x}\).

Note: f⁻¹(x) does NOT mean \(\color{blue}{\frac{1}{f(x)}}\). It is a completely separate function.

How to Find the Inverse of a Function

Step 1 — Replace f(x) with y

Rewrite the function as an equation in x and y.

Step 2 — Swap x and y

Exchange every x for y and every y for x in the equation.

Step 3 — Solve for y

Isolate the new y using algebra. The resulting expression is f⁻¹(x).

Example

Find the inverse of \(\color{blue}{f(x) = 3x – 6}\).

Step 1: \(\color{blue}{y = 3x – 6}\)
Step 2: \(\color{blue}{x = 3y – 6}\)
Step 3: \(\color{blue}{x + 6 = 3y}\) → \(\color{blue}{y = \frac{(x + 6)}{3}}\)
\(\color{blue}{f^{-1}(x) = \frac{(x + 6)}{3}}\)

Verify: f⁻¹\(\color{blue}{(f(2)) = f}\)⁻¹\(\color{blue}{(0) = \frac{(0 + 6)}{3} = 2}\) &checkmark;

Step-by-Step Summary

1. Write \(\color{blue}{y = f(x)}\).
2. Swap x and y: \(\color{blue}{x = f(y)}\).
3. Solve the resulting equation for y.
4. Write the answer as f⁻¹(x) = …
5. Verify by checking that f⁻¹\(\color{blue}{(f(a)) = a}\) for a test value a.

Watch: How to Find the Inverse of a Function (Video Lesson)

The Organic Chemistry Tutor walks through the swap-and-solve method with multiple examples:

Function Inverses – Worked Examples

Example 1: Find f⁻¹(x) for \(\color{blue}{f(x) = 3x – 6}\).

\(\color{blue}{y = 3x – 6}\) → swap: \(\color{blue}{x = 3y – 6}\) → \(\color{blue}{y = \frac{(x + 6)}{3}}\)
\(\color{blue}{f^{-1}(x) = \frac{(x + 6)}{3}}\)
Verify: \(\color{blue}{f(5) = 9}\), f⁻¹\(\color{blue}{(9) = \frac{15}{3} = 5}\) &checkmark;

Example 2: Find f⁻¹(x) for \(\color{blue}{f(x) = \frac{(x + 4)}{2}}\).

\(\color{blue}{y = \frac{(x + 4)}{2}}\) → swap: \(\color{blue}{x = \frac{(y + 4)}{2}}\) → \(\color{blue}{2x = y + 4}\) → \(\color{blue}{y = 2x – 4}\)
\(\color{blue}{f^{-1}(x) = 2x – 4}\)
Verify: \(\color{blue}{f(3) = \frac{7}{2} = 3.5}\), f⁻¹\(\color{blue}{(3.5) = 7 – 4 = 3}\) &checkmark;

Example 3: Find f⁻¹(x) for \(\color{blue}{f(x) = 2x + 4}\).

\(\color{blue}{y = 2x + 4}\) → swap: \(\color{blue}{x = 2y + 4}\) → \(\color{blue}{x – 4 = 2y}\) → \(\color{blue}{y = \frac{(x – 4)}{2}}\)
\(\color{blue}{f^{-1}(x) = \frac{(x – 4)}{2}}\)
Verify: \(\color{blue}{f(3) = 10}\), f⁻¹\(\color{blue}{(10) = \frac{6}{2} = 3}\) &checkmark;

Example 4: Find f⁻¹(x) for \(\color{blue}{f(x) = 5x + 10}\).

\(\color{blue}{y = 5x + 10}\) → swap: \(\color{blue}{x = 5y + 10}\) → \(\color{blue}{y = \frac{(x – 10)}{5}}\)
\(\color{blue}{f^{-1}(x) = \frac{(x – 10)}{5}}\)

More Practice: Introduction to Inverse Functions (Video Lesson)

This video from The Organic Chemistry Tutor provides an introduction to the concept of inverse functions and more worked examples:

Exercises for Function Inverses

Find the inverse of each function and verify your answer.

1. \(\color{blue}{f(x) = 4x + 8}\)
2. \(\color{blue}{f(x) = \frac{(x – 3)}{2}}\)
3. f(x) = −\(\color{blue}{x + 7}\)
4. \(\color{blue}{f(x) = 3x – 9}\)
5. \(\color{blue}{f(x) = \frac{(2x + 6)}{4}}\)

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Answers

1. f⁻¹(x) = \(\color{blue}{\frac{(x – 8)}{4}}\)
2. f⁻¹(x) = \(\color{blue}{2x + 3}\)
3. f⁻¹(x) = −\(\color{blue}{x + 7}\) (this function is its own inverse)
4. f⁻¹(x) = \(\color{blue}{\frac{(x + 9)}{3}}\)
5. f⁻¹(x) = \(\color{blue}{2x – 3}\)

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Free Function Inverses Worksheet

Ready to practice on your own? Download our free Function Inverses worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again — steady practice is the surest way to master Function Inverses before a quiz or test.

Download Inverse Functions Worksheet

Frequently Asked Questions

Does every function have an inverse?

No. Only one-to-one functions (where each output corresponds to exactly one input) have inverses. You can check this with the horizontal line test: if any horizontal line hits the graph more than once, the function does not have an inverse over its entire domain.

What is the difference between f⁻¹(x) and \(\color{blue}{\frac{1}{f(x)}}\)?

f⁻¹(x) is the inverse function, which undoes f. The notation \(\color{blue}{\frac{1}{f(x)}}\) is the reciprocal of f. They are generally not equal; for example, if \(\color{blue}{f(x) = 2x}\), then f⁻¹\(\color{blue}{(x) = \frac{x}{2}}\), while \(\color{blue}{\frac{1}{f(x)} = \frac{1}{(2x)}}\).

How can I verify that I found the correct inverse?

Substitute a specific value into f, then substitute that output into f⁻¹; you should get back the original input. Formally, f⁻¹\(\color{blue}{(f(a)) = a}\) for any a in the domain.

Related Topics

• Function Notation
• Composition of Functions
• Finding Domain and Range of a Function
• Adding and Subtracting Functions
• Increasing and Decreasing Functions