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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 first, then apply to the result". The order of operations for composite functions always moves from right to left, with the applied second.

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 .

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, $f g (x) \neq g f ($ .

Worked Example: Algebraic Substitution

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

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 .

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 $x$ .

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

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 $f$ .

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.

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.

In algebraic substitution, if you are squaring an inner expression like $3x$ , remember to square the entire term: $(3x)^2 = 9x^2$ , not .

A single function formed by the succession of two or more functions, where the second function acts on the result of the first.

The function that is applied first in a composite expression, located closest to the variable (e.g., $g$ in $fg(x)$ ).

The function that is applied second in a composite expression, acting upon the result of the inner function.

The mathematical process of associating each element of one set (the domain) with a unique element of another set (the range).

The set of all possible input values for which a given function is defined.

The set of all possible output values produced by a function.

The process of replacing a variable in an expression with another mathematical expression or value.