Lesson 4.2

Composition of Functions

Instead of adding or multiplying, you can feed one function's output into another. This is called composition: .

Introduction

Composition is the most powerful way to combine functions. Think of it as a machine pipeline: the output of one machine becomes the input of the next. Understanding composition is essential for inverse functions (Lessons 4.3–4.4).

Past Knowledge

Function notation, evaluating functions, and function operations (4.1).

Today's Goal

Evaluate and simplify and , understanding that order matters.

Future Success

Composition is the key to verifying inverse functions: .

Key Concepts

Notation

Read: = "f of g of x"

Means: — evaluate first, then plug that into

⚠ Order matters!

The Pipeline Analogy

Input

→

g(x)

Inner function

→

f(g(x))

Output

Worked Examples

Example 1: Evaluate at a Number

Basic

Let and . Find .

Inner first:

Then outer:

Example 2: Find a General Formula

Intermediate

Let and . Find and .

: Replace every x in f with g(x)

: Replace every x in g with f(x)

Notice: — order matters!

Example 3: Composition with Rational Functions

Advanced

Let and . Find .

Replace x in f with g(x)

Domain: →

Common Pitfalls

Wrong Order

means g first, then f. The inner function appears on the right in the notation but executes first.

Composition ≠ Multiplication

. Composition is substitution, not multiplication!

Real-Life Applications

A store offers 20% off, then you have a $10 coupon. The final price is a composition: Coupon(Discount(Price)). Changing the order (discount after coupon) gives a different result — composition order matters in daily life!