Decomposition of Functions — Definition, Formula & Examples

Decomposition of functions is the reverse of composition. You take one complicated function and break it into two or more simpler functions that, when composed, rebuild the original.

A decomposition of a function $h$ is a pair of functions $f$ and $g$ such that $h(x) = (f \circ g)(x) = f(g(x))$ for every $x$ in the domain of $h$ . Here $g$ is the inner function and $f$ is the outer function. Decompositions are not unique. The trivial choices $f(x) = x$ or $g(x) = x$ always work but are rarely useful.

Key Formula

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

Where:

$h$ = The original function you want to break apart
$g$ = The inner function, applied to x first
$f$ = The outer function, applied to the output of g

Worked Example

Problem: Decompose

h(x) = (3x + 2)^5

into an inner function

g

and an outer function

f

so that

h(x) = f(g(x))

Step 1: Ask what happens to

x

first. Before anything is raised to a power,

x

is multiplied by 3 and then 2 is added. That is the inner function.

g(x) = 3x + 2

Step 2: Ask what is done to that result. The whole quantity is raised to the 5th power. That is the outer function.

f(x) = x^5

Step 3: Verify by composing. Substituting

g(x)

into

f

gives back

h

. As a numeric check at

x = 1

, you get

g(1) = 5

and

f(5) = 3125

, which matches

h(1) = 5^5 = 3125

f(g(x)) = (3x + 2)^5 = h(x)

Answer:

g(x) = 3x + 2

(inner) and

f(x) = x^5

(outer), so

h(x) = f(g(x))

Why It Matters

In AP Calculus, the chain rule requires you to spot the inner and outer functions before you can differentiate expressions like

(3x+2)^5

\sin(x^2)

. The same skill drives u-substitution in integration, where the inner function becomes

u

Common Mistakes

Mistake: Swapping the inner and outer functions, for example writing

f(x) = 3x + 2

and

g(x) = x^5

Correction: With that choice,

f(g(x)) = 3x^5 + 2

, which is a different function. At

x = 1

it gives 5 instead of 3125. The inner function is always the operation applied to

x

first. Compose your answer to confirm it rebuilds

h

Related Terms

Function — the broader concept this term builds on
Argument of a Function — The inner function becomes the outer function's argument
Absolute Value Function — Common outer function, as in |x² − 9|
Continuous Function — Compositions of continuous functions stay continuous
Dependent Variable — Inner output feeds the outer function's input
Discontinuity — Inner or outer pieces can introduce breaks