Lesson 3.1

Arithmetic Operations on Functions

Just as you can add numbers, you can add entire functions. The catch? You have to make sure both functions "exist" at the same time.

Introduction

Prerequisite Connection: You know how to find the domain of a single function (Lesson 1.4). Now we are asking: if you have two machines, and , and you weld them together, what can you feed the new machine?

Today's Increment: We define as simply . The algebra is easy. The hard part—and the focus of this lesson—is finding the Combined Domain.

Why This Matters for Calculus: In Calculus, limits like require both functions to be defined near the point of interest. "Sum Rules" appear everywhere in Derivatives and Integrals.

Explanation of Key Concepts

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The Domain Intersection Rule

For any operation (add, subtract, multiply), the new function is defined ONLY where BOTH original functions are defined.

"The domain is the OVERLAP of the individual domains."

The Quotient Restriction

Numerator

Must be defined

Denominator

Must be defined

EXTRA RULE

Denominator

Worked Examples

Level: Intermediate

Example 1: Multiplying Roots

Let and . Find and its domain.

Step 1: Find Individual Domains

For : . Interval:
For : . Interval:

Step 2: Find Intersection

We need numbers that are GREATER than -2 AND LESS than 3.

Answer

Function:

Domain:

Level: Conceptual

Example 2: Adding Slopes (Graphical)

Visualize . We are adding a wave to a slanted line.

Notice how the blue line "wiggles" around the dotted gray line. We literally added the y-height of the sine wave to the y-height of the line at every point.

Level: Advanced

Example 3: Danger of Division

Let and . Find .

Step 1: Simplify

Critical Check

Even though it looks like the line

, the domain restriction from the original denominator (

) persists. That is why there is a hole!

Common Pitfalls

The "Cancellation" Trap:
If and , then . But the domain is NOT "all real numbers." Since was in the bottom, . The domain has a hole at 0.
Forgetting the "Intersection":
When adding and , students simplify to and stop. But verify the domain! AND . The ONLY number that works is 0. The domain is a single point!

Real-World Application

Economics: The Profit Function

Business relies on combining functions.

Revenue R(x): Money coming in (Price × Quantity).
Cost C(x): Money going out (Overhead + Materials).

Add them? No, we subtract them to find the most important function of all:
Profit P(x) = R(x) - C(x)