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. \(\color{blue}{f(x) = \frac{(2x + 6)}{4}}\)

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

What is an inverse function in math?

An inverse function undoes what the original function does. If f sends an input x to an output y, its inverse f⁻¹ sends y back to x. For example, if f(x) = x + 4, then f⁻¹(x) = x − 4, because adding 4 and then subtracting 4 returns the starting number. In symbols, f⁻¹(f(x)) = x and f(f⁻¹(x)) = x.

How do you find the inverse of a function?

Replace f(x) with y, swap x and y, and solve for y. For f(x) = 3x − 5, write y = 3x − 5, then swap to get x = 3y − 5. Add 5 to both sides and divide by 3: y = (x + 5)/3, so f⁻¹(x) = (x + 5)/3. Check it: f(2) = 1, and f⁻¹(1) = 6/3 = 2.

What does the graph of an inverse function look like?

The graph of an inverse is the original graph reflected over the line y = x. Every point (a, b) on f becomes (b, a) on f⁻¹. If f(x) = 2x + 3 passes through (2, 7), then its inverse passes through (7, 2). To sketch an inverse quickly, swap the coordinates of a few points and connect them.

Is an inverse function one-to-one?

Yes. A function has an inverse that is also a function only when it is one-to-one, meaning no two inputs share the same output, and the inverse is then one-to-one too. Use the horizontal line test: if any horizontal line crosses the graph more than once, there is no inverse function unless you restrict the domain. f(x) = 2x passes, but f(x) = x² fails on all real numbers.

How do you find the inverse of a quadratic function?

Restrict the domain first so the quadratic is one-to-one, then swap and solve. For f(x) = x² + 1 with x ≥ 0, write x = y² + 1, so y² = x − 1 and y = √(x − 1). Choose the positive root to match the restricted domain. The inverse is f⁻¹(x) = √(x − 1), defined for x ≥ 1.

What is inverse function notation?

The inverse of f is written f⁻¹(x) and read “f inverse of x.” The −1 is not an exponent here, so f⁻¹(x) does not mean 1/f(x). If g(x) = 2x, then g⁻¹(x) = x/2, while 1/g(x) = 1/(2x), a different function. Some books also write the inverse relationship as f⁻¹(b) = a whenever f(a) = b.

Related Topics

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