Defining Composition

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Idea

Composition feeds the output of one function into another. The inner function acts first, even though its name appears second in $f\circ g$ .

Visual guide

VISUAL GUIDEOrder changes the composite

f(g(x)) = (x + 1)²
g(f(x)) = x² + 1

With f(x) = x² and g(x) = x + 1, f(g(x)) shifts the parabola left, while g(f(x)) shifts it up. At x = 1 they give 4 and 2 respectively.

Method

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

Its domain contains exactly the inputs in the domain of $g$ whose outputs lie in the domain of $f$ . Both stages must be legal.

Worked example

Let $f(u)=\sqrt u$ and $g(x)=x-3$ . Then $(f\circ g)(x)=\sqrt{x-3}$ with domain $x\ge3$ . Reversing the order gives $(g\circ f)(x)=\sqrt x-3$ with domain $x\ge0$ . The different order changes both the rule and its domain.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

If f(x)=x² and g(x)=x+1, what is (f∘g)(2)?

Hint 1 · Find a starting point

Composition applies the inner function first.

Hint 2 · Take the next step

Compute g(2), then square that result.

Show the reasoning

Answer: 9

g(2)=3 and f(3)=9.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Common mistake

Composition is not multiplication. $f(g(x))$ generally differs from $f(x)g(x)$ .

Check your understanding

For $f(x)=1/x$ and $g(x)=x^2-4$ , find $f\circ g$ and its domain.

Show answer

$1/(x^2-4)$ , excluding $x=-2,2$ .

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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