SL 2.5—Composite functions, identity, finding inverse: IB DP Math AA Notes

What Are Composite Functions?

When you apply one function and then immediately apply another to the result, you create a composite function. Think of it like a two-step machine: the output of the first machine becomes the input of the second.

Composite Function

Given two functions $f (x)$ and $g (x)$ , the composite function $(f \circ g) (x)$ is defined as:
$(f \circ g) (x) = f (g (x))$
Read as " $f$ composed with $g$ " or " $f$ after $g$ ". The function $g$ is applied first, then $f$ is applied to that result.

The notation $(f \circ g) (x)$ can be a little confusing at first — notice that even though $f$ is written on the left, $g$ acts on $x$ first. A helpful way to remember: work from the inside out.

Analogy

Think of getting ready in the morning. Let $g$ = "put on socks" and $f$ = "put on shoes". The composition $f \circ g$ means: first put on socks, then put on shoes. Reversing the order ( $g \circ f$ ) gives a very different — and uncomfortable — result!

Working with Composite Functions

A key point about composition is that order matters — $f \circ g$ and $g \circ f$ are generally different functions.

Example

Example 1: Let $f (x) = x^{2}$ and $g (x) = x + 1$ . Find $(f \circ g) (x)$ and $(g \circ f) (x)$ .

Finding $(f \circ g) (x)$ :
$(f \circ g) (x) = f (g (x)) = f (x + 1) = (x + 1)^{2} = x^{2} + 2 x + 1$

Finding $(g \circ f) (x)$ :
$(g \circ f) (x) = g (f (x)) = g (x^{2}) = x^{2} + 1$

Clearly $x^{2} + 2 x + 1 \neq = x^{2} + 1$ , confirming that composition is not commutative in general.

Example

Example 2: Let $f (x) = 2 x - 1$ and $g (x) = x^{2} + 3$ . Evaluate $(f \circ g) (2)$ .

Step 1: Find $g (2)$ first:
$g (2) = 2^{2} + 3 = 4 + 3 = 7$

Step 2: Apply $f$ to this result:
$f (7) = 2 (7) - 1 = 14 - 1 = 13$

So $(f \circ g) (2) = 13$ .

Warning

A very common mistake is applying $f$ first and $g$ second when asked for $(f \circ g) (x)$ . Always remember: in $(f \circ g) (x) = f (g (x))$ , the rightmost function acts on $x$ first.

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What Are Composite Functions?

Working with Composite Functions

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