Lesson 4.4

Finding Inverse Functions

Now we find inverses algebraically: swap and , then solve for . This three-step process works for any one-to-one function.

Introduction

In Lesson 4.3, we understood what an inverse is. Now we learn how to find it. The process is always the same: replace with , swap and , and solve for . Then verify using composition.

Past Knowledge

Inverse concept (4.3), composition verification (4.2), solving equations.

Today's Goal

Find inverse functions algebraically and verify them using composition.

Future Success

Inverses of radical functions and exponentials build on this exact technique.

Key Concepts

The 3-Step Process

Replace f(x) with y.

Swap x and y.

Solve for y. That's f⁻¹(x).

Verifying with Composition

Check both compositions equal :

If both work, you've confirmed the inverse is correct.

Worked Examples

Example 1: Linear Function

Basic

Find the inverse of .

Replace f(x) with y, then swap

Solve for y

Example 2: Rational Function

Intermediate

Find the inverse of .

Swap x and y

Multiply both sides by , then collect y terms

Example 3: Verify with Composition

Verification

Verify that and are inverses.

Check

Both compositions equal — verified! ✓

Common Pitfalls

Forgetting to Swap

The most common mistake is solving for x without swapping first. You must swap x and y, then solve for y.

Skipping Verification

An algebra mistake in the middle will give a wrong inverse. Always verify with at least one composition.

Real-Life Applications

Encryption works by applying a function to a message; decryption is the inverse function. The RSA algorithm (HTTPS security) relies on the fact that some inverses are easy to compute if you know the key, but practically impossible without it.