AP®MathsCollege BoardPrecalculusRevision NotesExponential & Logarithmic FunctionsComposite & Inverse FunctionsConstructing Composite Functions

Constructing Composite Functions (College Board AP® Precalculus): Revision Note

Written by: Roger B

Reviewed by: Mark Curtis

Updated on 13 April 2026

Constructing composite functions

Why is function composition useful?

A composition of functions can be used to relate two quantities that are not directly related by an existing formula
E.g. if one function relates time to temperature
- and another relates temperature to pressure
- then composing them gives a function that relates time to pressure directly

How do I construct a composite function analytically?

When analytical (formula) representations of the functions $f$ and $g$ are available
- You can construct an analytical representation of $f (g (x))$
- by substituting $g (x)$ for every instance of $x$ in the formula for $f$
To do this
- Start by writing out the expression for $f (x)$
- Then replace every $x$ in that expression with the entire expression for $g (x)$
- Finally, simplify the resulting expression
E.g. if $f (x) = 3 x + 1$ and $g (x) = x^{2} - 4$
- Write out the expression for $f (x)$
  - $f (x) = 3 x + 1$
- Substitute the expression for $g (x)$ everywhere that $x$ appears
  - $f (g (x)) = 3 (x^{2} - 4) + 1$
- Expand the brackets and simplify
  - $f (g (x)) = 3 x^{2} - 12 + 1 = 3 x^{2} - 11$
Be careful to substitute $g (x)$ for every occurrence of $x$ in $f$
- E.g. if $f (x) = x^{2} + 2 x$ and $g (x) = x - 3$
  - $f (g (x)) = (x - 3)^{2} + 2 (x - 3) = x^{2} - 6 x + 9 + 2 x - 6 = x^{2} - 4 x + 3$
- Both the $x^{2}$ and $2 x$ terms required substitution

Examiner Tips and Tricks

When constructing $f (g (x))$ analytically, a common error is to only substitute $g (x)$ into one instance of $x$ in $f$ , while leaving other instances unchanged. Make sure you replace every $x$ in $f$ with $g (x)$ .

After substituting, simplify carefully, especially when fractions or squared expressions are involved. Show your working step by step to avoid algebraic errors.

How do you construct a composite function numerically or graphically?

A numerical representation (table) of $f \circ g$ can often be constructed by calculating pairs of values $(x, f (g (x)))$
- I.e. for each input $x$ , find $g (x)$ , then find $f$ at that value
- Record the pairs $(x, f (g (x)))$ to build a table for the composite function
A graphical representation of $f \circ g$ can be constructed similarly
- For selected input values $x$ , calculate or estimate $f (g (x))$
- Plot the resulting points $(x, f (g (x)))$
- Connect or sketch the graph based on the plotted points
These approaches are especially useful when one or both functions are given as tables or graphs rather than formulas

Examiner Tips and Tricks

Remember that $f (g (x))$ and $g (f (x))$ are generally different expressions. Always check which function is the outer function and which is the inner function.

Worked Example

The function $f$ is given by $f (x) = x^{2} - 2$ , and the function $g$ is given by $g (x) = \frac{(x + 1)}{x}$ . Which of the following is an expression for $f (g (x))$ ?

(A) $\frac{x^{3} + x^{2} - 2 x - 2}{x}$

(B) $\frac{x^{2} - 1}{x^{2} - 2}$

(D) $\frac{1 - x^{2}}{x^{2}}$

Answer:

To find $f (g (x))$ , substitute the expression for $g (x)$ everywhere that $x$ appears in the expression for $f (x)$

$f (g (x)) = {(\frac{(x + 1)}{x})}^{2} - 2$

Expand the brackets

$= \frac{{(x + 1)}^{2}}{x^{2}} - 2 = \frac{x^{2} + 2 x + 1}{x^{2}} - 2$

Write 2 as an equivalent fraction over $x^{2}$ , then combine the two fractions

$= \frac{x^{2} + 2 x + 1}{x^{2}} - \frac{2 x^{2}}{x^{2}} = \frac{x^{2} + 2 x + 1 - 2 x^{2}}{x^{2}} = \frac{2 x + 1 - x^{2}}{x^{2}} = \frac{1 + 2 x - x^{2}}{x^{2}}$

That is option C

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Constructing Composite Functions (College Board AP® Precalculus): Revision Note

Constructing composite functions

Why is function composition useful?

How do I construct a composite function analytically?

Examiner Tips and Tricks

How do you construct a composite function numerically or graphically?

Examiner Tips and Tricks

Worked Example

Unlock more, it's free!

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

Author: Roger B

Reviewer: Mark Curtis