Composition Of Functions

Function composition combines two functions by feeding the output of one into the input of another. Composition is central to differentiation via the Chain Rule.

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Notation	Expression	Meaning
Composite Function	$(f \circ g) (x)$	$f (g (x))$ : evaluate g first, then f
Composite Domain	$Dom (f \circ g)$	${x \in Dom (g) ∣ g (x) \in Dom (f)}$
Commutativity	In general, $f \circ g \neq g \circ f$	Order of composition matters

Definition of Composition

Given two functions f and g, the composite function $f \circ g$ (also called the composition of f and g) is defined by:

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

The domain of $f \circ g$ is the set of all x in the domain of g such that $g (x)$ is in the domain of f:

Dom (f \circ g) = {x \in Dom (g) ∣ g (x) \in Dom (f)}

Let $f (x) = \sqrt{x}$ and $g (x) = x + 1$ . Find $(f \circ g) (x)$ and $(g \circ f) (x)$ , and determine their domains.

Solution

$(f \circ g) (x) = f (g (x)) = f (x + 1) = \sqrt{x + 1}$ .
Domain: $x + 1 \geq 0 ⟹ x \geq - 1$ , so $Dom (f \circ g) = [- 1, + \infty)$ .
$(g \circ f) (x) = g (f (x)) = g (\sqrt{x}) = \sqrt{x} + 1$ .
Domain: $x \geq 0$ , so $Dom (g \circ f) = [0, + \infty)$ .

Notice that $(f \circ g) (x) \neq (g \circ f) (x)$ in general.

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Definition of Composition

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