Lesson 4.1

Function Operations

You can add, subtract, multiply, and divide functions just like numbers — the result is a brand-new function.

Introduction

Just as you can add two numbers, you can add two functions: . The same goes for subtraction, multiplication, and division. This is the gateway to function composition (Lesson 4.2) and inverse functions (Lessons 4.3–4.4).

Past Knowledge

Function notation and evaluating functions from earlier algebra.

Today's Goal

Perform addition, subtraction, multiplication, and division of functions and state the resulting domain.

Future Success

Lesson 4.2 introduces composition — a more powerful way to combine functions.

Key Concepts

The Four Operations

Domain of Combined Functions

For :

Domain = (domain of f) ∩ (domain of g)

For :

Domain = (domain of f) ∩ (domain of g), excluding where g(x) = 0

Worked Examples

Example 1: Sum and Difference

Basic

Let and . Find and .

Sum

Difference

Domain: all real numbers (both are polynomials)

Example 2: Product

Intermediate

Let and . Find and evaluate at .

Multiply

Evaluate at x = 3

Example 3: Quotient with Domain Restriction

Advanced

Let and . Find .

Divide and simplify

State domain restriction

when , so the domain excludes 3.

Common Pitfalls

Dropping the Negative in (f − g)

When subtracting, distribute the negative: . You must subtract every term of g(x).

Forgetting Quotient Domain

Even after simplification, the original domain restriction remains. If , then forever.

Real-Life Applications

In business, profit = revenue − cost: . This is function subtraction in action! Every time you combine two real-world models, you're performing function operations.