PrecalculusPrecalculusHigh school

Composition of Functions

Build f(g(x)) and find its domain.

Composing functions means feeding the output of one into another: $(f \circ g)(x) = f(g(x))$ . The inner function $g$ acts first. Composition models processes that happen in stages, such as converting currency and then applying a tax, and it is the structure the chain rule in calculus differentiates.

Order matters: in general $f(g(x)) \ne g(f(x))$ . The domain of $f \circ g$ contains the inputs $x$ that are in the domain of $g$ and whose output $g(x)$ is in the domain of $f$ , so both layers can restrict it.

Key ideas

Evaluate from the inside out: compute $g(x)$ , then apply $f$ .
To build a formula, replace every $x$ in $f$ with the whole expression $g(x)$ , in parentheses.
Decomposing a function, such as writing $\sqrt{x^2 + 1}$ as $f(g(x))$ , is the reverse skill.
Check the domain of the inner function and of the result.

Worked example

Let $f(x) = x^2 - 1$ and $g(x) = 2x + 3$ . Find $f(g(x))$ and $g(f(x))$ .

$f(g(x)) = (2x + 3)^2 - 1 = 4x^2 + 12x + 8$ .
$g(f(x)) = 2(x^2 - 1) + 3 = 2x^2 + 1$ .
The two results differ, showing order matters.
Spot check at $x = 0$ : $f(g(0)) = f(3) = 8$ and $g(f(0)) = g(-1) = 1$ .

Answer: $f(g(x)) = 4x^2 + 12x + 8$ , $g(f(x)) = 2x^2 + 1$

Common mistakes

Multiplying $f(x) \cdot g(x)$ instead of composing.
Applying the outer function first.
Dropping parentheses when substituting, as in $2x + 3^2$ .

Practice questions

Try each one, then open the answer. Numerical answers on this page are verified by an automated calculation check.

1.With $f(x) = 3x$ and $g(x) = x + 4$ , find $f(g(2))$ .
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$18$
2.With the same functions, find $g(f(2))$ .
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$10$
3.With $f(x) = \sqrt{x}$ and $g(x) = x^2 + 9$ , find $f(g(4))$ .
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$5$

Keep practising

Calculators

Function Composition Calculator