Function Multiplication- Techniques and Examples Explained

What Is Function Multiplication?

Function multiplication is exactly what it sounds like. You take two functions and multiply them together to get a new function. If you have f(x) and g(x), then f(x) · g(x) gives you a combined output based on both inputs.

You do this constantly in algebra without calling it by the fancy name. When you expand (x+2)(x-3), you're multiplying two functions. The difference is that in function multiplication, you're working with named entities that can represent more than just simple polynomials.

The result is called a product function, written as (f · g)(x) or f(x)g(x). That new function then accepts the same input as the originals and spits out their multiplied values.

The Core Technique

Here's the rule: to multiply functions, multiply the outputs, not the inputs.

That's it. That's the whole concept.

Given f(x) and g(x), the product function h(x) = f(x) · g(x) means you evaluate both functions at x, then multiply those results. You're not combining the x-values. You're combining the y-values.

Step-by-Step Process

• Identify your two functions
• Write down what each function equals at a given x
• Multiply those two expressions together
• Simplify the result

Examples That Actually Make Sense

Example 1: Simple Polynomials

Let f(x) = x + 1 and g(x) = x - 2

Find (f · g)(x):

Step 1: Write out the multiplication

(x + 1)(x - 2)

Step 2: Expand using FOIL or distributive property

= x² - 2x + x - 2

Step 3: Simplify

= x² - x - 2

So (f · g)(x) = x² - x - 2. Test it: let x = 3. f(3) = 4, g(3) = 1. 4 × 1 = 4. Does x² - x - 2 equal 4 when x = 3? 9 - 3 - 2 = 4. Yes.

Example 2: Quadratic Times Linear

Let f(x) = x² and g(x) = 2x + 3

Find (f · g)(x):

(x²)(2x + 3) = 2x³ + 3x²

Nothing fancy. You multiply each term in the second function by the entire first function.

Example 3: Trig Functions

Let f(x) = sin(x) and g(x) = cos(x)

Find (f · g)(x):

(f · g)(x) = sin(x) · cos(x) = ½ sin(2x)

This is a product-to-sum identity. The simplified form isn't always required—sometimes the product form is exactly what you need.

Example 4: Nested Function Multiplication

Let f(x) = x² + 1 and g(x) = f(x) · f(x)

This is squaring a function. g(x) = (x² + 1)²

Expand: (x² + 1)(x² + 1) = x⁴ + 2x² + 1

How To: Getting Started

Here's how you actually multiply functions in practice:

Method 1: Direct Multiplication

Multiply the expressions term-by-term.

f(x) = 3x + 2

g(x) = 4x - 1

(f · g)(x) = (3x + 2)(4x - 1)

= 12x² - 3x + 8x - 2

= 12x² + 5x - 2

Method 2: Composition Then Evaluate

Sometimes you multiply after substituting one function into another.

f(x) = x²

g(x) = x + 1

f(g(x)) = (x + 1)²

g(f(x)) = x² + 1

These are different. f(g(x)) ≠ g(f(x)) in general. You're not multiplying the functions here—you're composing them. Don't confuse composition with multiplication.

Method 3: Numerical Evaluation

Find (f · g)(2) if f(x) = x + 5 and g(x) = 3x.

f(2) = 7

g(2) = 6

(f · g)(2) = 7 × 6 = 42

Sometimes you just need a number, not a formula.

Common Mistakes That Will Cost You Points

When Each Method Makes Sense

The Bottom Line

Function multiplication is basic arithmetic applied to function outputs. You evaluate each function separately at the same input, then multiply those results. That's the entire process.

Where people get tripped up is confusing it with composition, forgetting to distribute properly, or trying to apply fancy techniques when simple expansion works fine.

Master the basics first. Expand correctly. Test with numbers. You'll get there.