Composite Functions

The Sequence of Functions

Every time you put on your socks and then your shoes, you are completing a sequence of operations where the order changes the outcome. In mathematics, a composite function is formed when the output of one function becomes the input of another. It is essentially a "function of a function".

In the notation $fg(x)$ , the function closest to the variable $x$ is the inner function and is applied first. Therefore, $fg(x)$ means "apply $g$ first, then apply $f$ to the result". The order of operations for composite functions always moves from right to left, with the outer function applied second.

You may also see $fg(x)$ written identically as $f(g(x))$ . This nested notation explicitly shows that the entire expression for $g(x)$ is substituted into function $f$ .

Mapping and Function Machines

Visualising this process helps track how values change. A composite mapping diagram for $fg(x)$ involves three sets of numbers.

The initial input values form the domain. These inputs go into $g(x)$ , producing intermediate outputs, which then serve as inputs for $f(x)$ to produce the final outputs in the range.

We can also represent this using function machines, where the output wire of machine $G$ connects directly to the input of machine $F$ . If $x \xrightarrow{g} y$ and $y \xrightarrow{f} z$ , then the successive mapping is $x \xrightarrow{fg} z$ .

Evaluating Composite Functions Numerically

When given a specific number to substitute, calculate the value step-by-step. Start by evaluating the inner function, then substitute that numerical result directly into the outer function.

Worked Example: Numerical Evaluation

If $f(x) = 3x^2 + 1$ and $g(x) = \frac{4}{x^2}$ , find $gf(1)$ .

Step 1: Calculate the inner function, $f(1)$ .

$f(1) = 3(1)^2 + 1 = 3 + 1 = 4$

Step 2: Use this result as the input for the outer function, $g(x)$ .

$g(4) = \frac{4}{4^2}$

Step 3: Calculate the final answer.

$g(4) = \frac{4}{16} = 0.25$

Algebraic Substitution

To find a general formula for $fg(x)$ , you must perform algebraic substitution, which involves replacing a variable in an expression with another entire mathematical expression.

Write out the expression for the outer function, replacing every $x$ with empty brackets. Then, "drop" the entire expression for the inner function into those brackets and expand to simplify.

Sometimes you will need to find a self-composition like $ff(x)$ (also written as $f^2(x)$ ). This means applying the function $f$ to its own output, which is not the same as squaring the original function. Furthermore, composite functions are non-commutative; in almost all cases, $fg(x) \neq gf(x)$ .

Worked Example: Algebraic Substitution

Given $f(x) = 2x - 1$ and $g(x) = x^2 + 4$ , find an expression for $fg(x)$ .

Step 1: Write the outer function, $f(x)$ , with empty brackets replacing $x$ .

$2(\dots) - 1$

Step 2: Substitute the entire inner function, $g(x)$ , into the brackets.

$2(x^2 + 4) - 1$

Step 3: Expand the brackets and simplify the expression.

$2x^2 + 8 - 1$
$fg(x) = 2x^2 + 7$

Solving Equations with Composite Functions

Higher-tier Edexcel papers often require you to solve equations in the form $fg(x) = k$ . To solve these, find the algebraic expression for the composite function first, set it equal to $k$ , and solve the resulting equation.

Worked Example: Solving an Equation

Given $f(x) = 3x + 4$ and $g(x) = \frac{\sqrt{x} + 2}{5}$ , solve $gf(x) = 3$ .

Step 1: Find the algebraic expression for $gf(x)$ by substituting $f(x)$ into $g(x)$ .

$gf(x) = \frac{\sqrt{3x + 4} + 2}{5}$

Step 2: Set the composite expression equal to $3$ .

$\frac{\sqrt{3x + 4} + 2}{5} = 3$

Step 3: Solve the equation to isolate $x$ .

$\sqrt{3x + 4} + 2 = 15$
$\sqrt{3x + 4} = 13$
$3x + 4 = 169$
$3x = 165$
$x = 55$

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Exam Tips

Common Mistake
Students frequently fall into the "Left-to-Right Trap" by applying $f$ then $g$ for $fg(x)$ — always remember that the letter nearest the $x$ goes first, so you apply $g$ first, then $f$ .
2
Do not confuse $fg(x)$ with multiplying the two functions together ( $f(x) \times g(x)$ ); treating composition as multiplication will result in zero marks.
3
Examiners will award method marks (M1) for showing the correct algebraic substitution of one function into another, even if you make an arithmetic error when expanding or simplifying later.
4
In algebraic substitution, if you are squaring an inner expression like $3x$ , remember to square the entire term: $(3x)^2 = 9x^2$ , not $3x^2$ .

Key Terms(7)

Composite function: A single function formed by the succession of two or more functions, where the second function acts on the result of the first.
Inner function: The function that is applied first in a composite expression, located closest to the variable (e.g., $g$ in $fg(x)$ ).
Outer function: The function that is applied second in a composite expression, acting upon the result of the inner function.
Mapping: The mathematical process of associating each element of one set (the domain) with a unique element of another set (the range).
Domain: The set of all possible input values for which a given function is defined.
Range: The set of all possible output values produced by a function.
Algebraic substitution: The process of replacing a variable in an expression with another mathematical expression or value.

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Key Terms(7)

Composite function: A single function formed by the succession of two or more functions, where the second function acts on the result of the first.
Inner function: The function that is applied first in a composite expression, located closest to the variable (e.g., $g$ in $fg(x)$ ).
Outer function: The function that is applied second in a composite expression, acting upon the result of the inner function.
Mapping: The mathematical process of associating each element of one set (the domain) with a unique element of another set (the range).
Domain: The set of all possible input values for which a given function is defined.
Range: The set of all possible output values produced by a function.
Algebraic substitution: The process of replacing a variable in an expression with another mathematical expression or value.

Composite Functions

The Sequence of Functions

You may also see $fg(x)$ written identically as $f(g(x))$ . This nested notation explicitly shows that the entire expression for $g(x)$ is substituted into function $f$ .

Mapping and Function Machines

Visualising this process helps track how values change. A composite mapping diagram for $fg(x)$ involves three sets of numbers.

The initial input values form the domain. These inputs go into $g(x)$ , producing intermediate outputs, which then serve as inputs for $f(x)$ to produce the final outputs in the range.

Evaluating Composite Functions Numerically

When given a specific number to substitute, calculate the value step-by-step. Start by evaluating the inner function, then substitute that numerical result directly into the outer function.

Worked Example: Numerical Evaluation

If $f(x) = 3x^2 + 1$ and $g(x) = \frac{4}{x^2}$ , find $gf(1)$ .

Step 1: Calculate the inner function, $f(1)$ .

$f(1) = 3(1)^2 + 1 = 3 + 1 = 4$

Step 2: Use this result as the input for the outer function, $g(x)$ .

$g(4) = \frac{4}{4^2}$

Step 3: Calculate the final answer.

$g(4) = \frac{4}{16} = 0.25$

Algebraic Substitution

To find a general formula for $fg(x)$ , you must perform algebraic substitution, which involves replacing a variable in an expression with another entire mathematical expression.

Write out the expression for the outer function, replacing every $x$ with empty brackets. Then, "drop" the entire expression for the inner function into those brackets and expand to simplify.

Worked Example: Algebraic Substitution

Given $f(x) = 2x - 1$ and $g(x) = x^2 + 4$ , find an expression for $fg(x)$ .

Step 1: Write the outer function, $f(x)$ , with empty brackets replacing $x$ .

$2(\dots) - 1$

Step 2: Substitute the entire inner function, $g(x)$ , into the brackets.

$2(x^2 + 4) - 1$

Step 3: Expand the brackets and simplify the expression.

$2x^2 + 8 - 1$
$fg(x) = 2x^2 + 7$

Solving Equations with Composite Functions

Worked Example: Solving an Equation

Given $f(x) = 3x + 4$ and $g(x) = \frac{\sqrt{x} + 2}{5}$ , solve $gf(x) = 3$ .

Step 1: Find the algebraic expression for $gf(x)$ by substituting $f(x)$ into $g(x)$ .

$gf(x) = \frac{\sqrt{3x + 4} + 2}{5}$

Step 2: Set the composite expression equal to $3$ .

$\frac{\sqrt{3x + 4} + 2}{5} = 3$

Step 3: Solve the equation to isolate $x$ .

$\sqrt{3x + 4} + 2 = 15$
$\sqrt{3x + 4} = 13$
$3x + 4 = 169$
$3x = 165$
$x = 55$

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students frequently fall into the "Left-to-Right Trap" by applying $f$ then $g$ for $fg(x)$ — always remember that the letter nearest the $x$ goes first, so you apply $g$ first, then $f$ .
2
Do not confuse $fg(x)$ with multiplying the two functions together ( $f(x) \times g(x)$ ); treating composition as multiplication will result in zero marks.
3
Examiners will award method marks (M1) for showing the correct algebraic substitution of one function into another, even if you make an arithmetic error when expanding or simplifying later.
4
In algebraic substitution, if you are squaring an inner expression like $3x$ , remember to square the entire term: $(3x)^2 = 9x^2$ , not $3x^2$ .

Key Terms(7)

Composite function: A single function formed by the succession of two or more functions, where the second function acts on the result of the first.
Inner function: The function that is applied first in a composite expression, located closest to the variable (e.g., $g$ in $fg(x)$ ).
Outer function: The function that is applied second in a composite expression, acting upon the result of the inner function.
Mapping: The mathematical process of associating each element of one set (the domain) with a unique element of another set (the range).
Domain: The set of all possible input values for which a given function is defined.
Range: The set of all possible output values produced by a function.
Algebraic substitution: The process of replacing a variable in an expression with another mathematical expression or value.

Previous NoteInverse Functions Next Topic: Graphical MethodsCoordinates

Back to Functions

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(7)

Composite function: A single function formed by the succession of two or more functions, where the second function acts on the result of the first.
Inner function: The function that is applied first in a composite expression, located closest to the variable (e.g., $g$ in $fg(x)$ ).
Outer function: The function that is applied second in a composite expression, acting upon the result of the inner function.
Mapping: The mathematical process of associating each element of one set (the domain) with a unique element of another set (the range).
Domain: The set of all possible input values for which a given function is defined.
Range: The set of all possible output values produced by a function.
Algebraic substitution: The process of replacing a variable in an expression with another mathematical expression or value.