Function Composition

What is Composition?

Composition is feeding the output of one function into another.

$f(g(x))$

First evaluate $g(x)$ , then feed that result into $f$ .

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

Read as “f composed with g” or “f of g of x”.

Think of it as a chain: input → $g$ → output → $f$ → final output

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

Find $(f \circ g)(x)$ :

\begin{aligned} (f \circ g)(x) &= f(g(x)) \\ &= f(x + 3) \\ &= (x + 3)^2 \end{aligned}

Find $(g \circ f)(x)$ :

\begin{aligned} (g \circ f)(x) &= g(f(x)) \\ &= g(x^2) \\ &= x^2 + 3 \end{aligned}

Notice that $(x + 3)^2 \neq x^2 + 3$ .

Composition is not commutative. In general, $f \circ g \neq g \circ f$ .

Find $(f \circ g)(2)$ where $f(x) = x^2$ and $g(x) = x + 3$ :

Method 1: Work from the inside out

\begin{aligned} g(2) &= 2 + 3 = 5 \\ f(g(2)) &= f(5) = 25 \end{aligned}

Method 2: Find the composed function first

\begin{aligned} (f \circ g)(x) &= (x + 3)^2 \\ (f \circ g)(2) &= (2 + 3)^2 = 25 \end{aligned}

Sometimes you need to work backwards: express a function as a composition.

Example: Write $h(x) = \sqrt{x^2 + 1}$ as $f(g(x))$ .

Think: what’s the “outer” operation and what’s the “inner” operation?

So: $f(x) = \sqrt{x}$ and $g(x) = x^2 + 1$

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

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

Example: $f(x) = \sqrt{x}$ and $g(x) = x - 4$

For $f(g(x)) = \sqrt{x - 4}$ :

Domain of $f \circ g$ : $x \geq 4$