AP®MathsCollege BoardPrecalculusRevision NotesExponential & Logarithmic FunctionsComposite & Inverse FunctionsDecomposing Functions

Decomposing Functions (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 13 April 2026

Decomposing functions

What does it mean to decompose a function?

Functions given analytically can often be decomposed (broken down) into less complicated functions
- This means rewriting a single function
  - as the composition of two or more simpler functions
- E.g. the function $h (x) = (2 x + 1)^{3}$ can be decomposed as
  - $h (x) = f (g (x))$
    - where $f (x) = x^{3}$
    - and $g (x) = 2 x + 1$
You can check a decomposition is correct by substituting the inner function back into the outer function
- If every $x$ in the outer function gets replaced by the entire inner function
  - then the result should match the original
- In the example above
  - $f (x) = x^{3}$ and $g (x) = 2 x + 1$
  - so $f (g (x)) = (2 x + 1)^{3}$
  - this matches $h (x)$ ✓

How do you find a decomposition?

Look for a structure where one operation is applied to the result of another
- Start by identifying an "inner" part of the expression
  - this becomes $g (x)$
- To find the outer function $f$
  - Take the original expression and swap out every instance of the inner part for a plain $x$
  - What remains is $f (x)$
E.g. decompose $h (x) = \sqrt{x^{2} - 5}$
- The inner part is $x^{2} - 5$
  - so let $g (x) = x^{2} - 5$
- The outer operation is taking the square root
  - so let $f (x) = \sqrt{x}$
- Then $h (x) = f (g (x)) = \sqrt{x^{2} - 5}$ ✓
Decompositions are not unique
- The same function can sometimes be decomposed in different ways
- E.g. $h (x) = (x + 3)^{2} + 5$ could be decomposed as $f (g (x))$
  - with $f (x) = x^{2} + 5$ and $g (x) = x + 3$
  - or with $f (x) = x + 5$ and $g (x) = (x + 3)^{2}$

Vertical and horizontal translations of a function $f$ correspond to additive transformations
- These can be understood as compositions of $f$ with $g (x) = x + k$
  - $f (g (x)) = f (x + k)$
    - That is a horizontal translation by $k$ units (to the left if $k$ is positive, to the right if $k$ is negative)
  - $g (f (x)) = f (x) + k$
    - That is a vertical translation up by $k$ units (up if $k$ is positive, down if $k$ is negative)
Vertical and horizontal dilations of a function $f$ correspond to multiplicative transformations
- These can be understood as compositions of $f$ with $g (x) = k x$
  - $f (g (x)) = f (k x)$
    - That is a horizontal dilation by a factor of $\frac{1}{k}$
  - $g (f (x)) = k f (x)$
    - That is a vertical dilation by a factor of $k$
This means that every transformation you have studied can be thought of as a composition of functions

Worked Example

The function $h$ is given by $h (x) = 3 (x + 4)^{2} - 7$ .

(a) Write $h (x)$ as a composition $h (x) = p (q (x))$ , where $q$ is a linear function. Identify $p (x)$ and $q (x)$ .

Answer:

The linear inner part of the expression is $x + 4$

so that should be $q (x)$

To find $p (x)$ replace $x + 4$ with plain $x$ everywhere that it appears in $h (x)$

Let $q (x) = x + 4$ and $p (x) = 3 x^{2} - 7$

Then $p (q (x)) = 3 (x + 4)^{2} - 7 = h (x)$

(b) Describe the graph of $h$ as a transformation of the graph of $p$ .

Answer:

$p$ has been composed with $q (x) = x + 4$

$p (q (x)) = p (x + 4)$
That is an additive transformation of $p (x)$ , corresponding to a horizontal translation

The graph of $h$ is a horizontal translation of
the graph of $p (x)$ by 4 units to the left

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Decomposing Functions (College Board AP® Precalculus): Revision Note

Decomposing functions

What does it mean to decompose a function?

How do you find a decomposition?

Worked Example

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis

Decomposing Functions (College Board AP® Precalculus): Revision Note

Decomposing functions

What does it mean to decompose a function?

How do you find a decomposition?

How are transformations related to compositions?

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Roger B

Reviewer: Mark Curtis