Function Operations- Addition, Subtraction, Multiplication

What Are Function Operations?

Function operations let you combine, subtract, and multiply functions just like you do with regular numbers. Instead of working with isolated expressions, you're working with entire functions that map inputs to outputs.

This is fundamental for anyone studying algebra, calculus, or any math that builds on functions. Once you understand the mechanics, the process is straightforward.

Function Addition

Adding functions means you add their outputs for any given input. If you have f(x) and g(x), then (f + g)(x) = f(x) + g(x).

That's it. You evaluate both functions at x, then add the results.

Example

Let f(x) = 3x + 2 and g(x) = x² - 1.

(f + g)(x) = (3x + 2) + (x² - 1) = x² + 3x + 1

The resulting function combines both expressions. You can verify this by plugging in a value—say x = 2:

f(2) = 3(2) + 2 = 8
g(2) = 2² - 1 = 3
(f + g)(2) = 8 + 3 = 11

And from the combined function: x² + 3x + 1 = 4 + 6 + 1 = 11 ✓

Function Subtraction

Subtraction works the same way. (f - g)(x) = f(x) - g(x). Watch the signs carefully—subtracting a polynomial means you subtract every term.

Example

Using the same functions: f(x) = 3x + 2 and g(x) = x² - 1

(f - g)(x) = (3x + 2) - (x² - 1) = 3x + 2 - x² + 1 = -x² + 3x + 3

Notice how the minus sign flips both terms in g(x). This is where students make mistakes—always distribute the negative sign to every term.

Function Multiplication

Multiplication is where it gets more involved. (f · g)(x) = f(x) · g(x). You're multiplying the entire expressions together, which often produces higher-degree polynomials.

Example

Same functions: f(x) = 3x + 2 and g(x) = x² - 1

(f · g)(x) = (3x + 2)(x² - 1)

Use distribution or FOIL:

= 3x(x² - 1) + 2(x² - 1)
= 3x³ - 3x + 2x² - 2
= 3x³ + 2x² - 3x - 2

Check at x = 1:

f(1) · g(1) = (3 + 2)(1 - 1) = 5 · 0 = 0

Combined function: 3(1)³ + 2(1)² - 3(1) - 2 = 3 + 2 - 3 - 2 = 0 ✓

Domain Considerations

After any operation, the resulting function's domain is the intersection of the original domains. If f(x) has domain all real numbers but g(x) = 1/x, then (f + g)(x) has domain all real numbers except x = 0.

Don't ignore this. Problems on tests often ask for domain restrictions after operations.

Quick Reference Table

How to Perform Function Operations

Follow these steps in order:

1. Identify the functions — Write down both f(x) and g(x) clearly.
2. Set up the operation — Replace f(x) and g(x) with their expressions.
3. Combine like terms — Add, subtract, or multiply the expressions.
4. Simplify — Put the result in standard form (descending powers).
5. Check the domain — Note any restrictions from the original functions.

Practice Problem

Given f(x) = 2x² - 3x and g(x) = x + 4, find (f + g)(x) and (f · g)(x).

Solution:

(f + g)(x) = (2x² - 3x) + (x + 4) = 2x² - 2x + 4

(f · g)(x) = (2x² - 3x)(x + 4) = 2x²(x + 4) - 3x(x + 4) = 2x³ + 8x² - 3x² - 12x = 2x³ + 5x² - 12x

Common Mistakes to Avoid

• Forgetting to distribute negative signs when subtracting
• Multiplying only the first terms together instead of every term
• Ignoring domain restrictions after operations
• Combining unlike terms (you can't add x² and x together)

When You'll Use This

Function operations appear in:

• Calculus — composite functions, derivatives of sums and products
• Physics — combining displacement functions to find total motion
• Economics — adding cost and revenue functions to find profit
• Computer science — composing transformations in graphics

The mechanics stay the same regardless of the application. Master the basics here and everything builds cleanly on top.