Function Operations and Composition

Functions can be combined in several ways.

Basic Operations

Given $f(x)$ and $g(x)$:

Addition

$(f + g)(x) = f(x) + g(x)$

Subtraction

$(f - g)(x) = f(x) - g(x)$

Multiplication

$(f \cdot g)(x) = f(x) \cdot g(x)$

Division

$(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, where $g(x) \neq 0$

Example

Let $f(x) = x^2$ and $g(x) = x + 1$

• $(f + g)(x) = x^2 + x + 1$
• $(f - g)(x) = x^2 - x - 1$
• $(f \cdot g)(x) = x^2(x + 1) = x^3 + x^2$
• $(\frac{f}{g})(x) = \frac{x^2}{x + 1}$

Function Composition

Composition means plugging one function into another.

$(f \circ g)(x) = f(g(x))$

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

Example

Let $f(x) = x^2$ and $g(x) = x + 3$

$(f \circ g)(x) = f(g(x)) = f(x + 3) = (x + 3)^2$

$(g \circ f)(x) = g(f(x)) = g(x^2) = x^2 + 3$

Note: $f \circ g \neq g \circ f$ in general!

Decomposing Functions

Given $h(x) = \sqrt{x^2 + 1}$

We can write $h = f \circ g$ where:

• $g(x) = x^2 + 1$ (inner function)
• $f(x) = \sqrt{x}$ (outer function)

Then $f(g(x)) = \sqrt{x^2 + 1} = h(x)$ ✓