Lesson 3.2

Composition of Functions

Functions inside functions. We call this "The Chain." It's the most powerful operation in calculus, forming the backbone of the Chain Rule.

Introduction

Prerequisite Connection: You've seen . That was actually a composition! You composed the function with the linear expression .

Today's Increment: We formalize this process as . We take the output of one machine and dump it directly into the hopper of the next machine.

Why This Matters for Calculus: 90% of derivatives you calculate will require the Chain Rule, which tells you how to differentiate composite functions. If you can't identify the "inner" and "outer" layers, you cannot pass Calculus.

Explanation of Key Concepts

The Composition Notation

(f O g)(x) = f(g(x))

Read it as "f composed with g" or "f of g of x."

Right to Left: We evaluate first (the inside).
Feed Forward: The output of becomes the input of .

Input xg(x)

Input g(x)f(g(x))

Worked Examples

Level: Basic

Example 1: Numeric Evaluation

Given and , find .

Step 1: Inside First

Step 2: Outside Second

Plug the result (5) into f.

Answer

Level: Intermediate

Example 2: Order Matters

Using and , verify that .

f(g(x)): Plug g into f

Result: Parabola shifted RIGHT 2.

g(f(x)): Plug f into g

Result: Parabola shifted DOWN 2.

Conclusion: They are completely different graphs! Composition is NOT commutative.

Level: Advanced (Calculus Prep)

Example 3: Decomposition (The "Chain Rule" Skill)

Express as .

To differentiate this in calculus, you must identify the "Inner" and "Outer" functions.

Strategy: Order of Operations

If you plugged in a number for x, what would you do first?

First, I would calculate . (This is the INNER function, g).
Then, I would take the square root of that result. (This is the OUTER function, f).

Solution

Common Pitfalls

Mixing order:
means g goes FIRST. It's counter-intuitive because we read left-to-right, but evaluation happens from the inside out. Remember: "Circle means Of". f circle g is "f OF g".
Domain Errors:
Finding the domain of is tricky. must be allowed in , AND the output must be allowed in . You have to check the domain at every step of the chain.

Real-World Application

Currency Exchange and Fees

Imagine you are sending money abroad.

converts Dollars to Euros.
takes a 2% fee from the Euros.

The final amount you receive is . You convert FIRST, then pay the fee. If you did (pay fee in dollars first, then convert), you might actually get a different amount if the fee structure is nonlinear!