Composite functions

By the end you can form and evaluate composite functions such as $gf(x)$ and $f^2(x)$ , decide whether a composite exists, and find its range in simple cases.

Socks then shoes: normal. Shoes then socks: a fashion statement. Same two actions, different order, different result.

Functions work the same way. A composite function feeds the output of one function straight into another.

Two words you'll need:

The domain of a function is the set of inputs it's allowed to take. " $x \in \mathbb{R}$ " means " $x$ can be any real number" ( $\in$ means "belongs to", $\mathbb{R}$ is the set of all real numbers).
The range is the set of outputs it actually produces.
In the diagram, $x \mapsto x + 3$ is read " $x$ is sent to $x + 3$ ". It's just another way of writing $f(x) = x + 3$ .

The composite function $gf$ is defined by $gf(x) = g(f(x))$ : apply $f$ first, then apply $g$ to the result. It can only be formed when the range of $f$ lies within the domain of $g$ .

$f(x) = 2x - 1$ and $g(x) = x^2 + 3$ , both for $x \in \mathbb{R}$ . Find $gf(x)$ , $fg(x)$ and $f^2(x)$ .

What does $gf(x)$ mean?

$f(x) = 3x + 1$ and $g(x) = x^2 - 2$ , for $x \in \mathbb{R}$ . Find $gf(2)$ .

$f(x) = 2x + 5$ and $g(x) = x^2 - 1$ , for $x \in \mathbb{R}$ . Find and simplify $fg(x)$ .

Socks, then shoes