Use function notation: form composite functions, find inverse functions, state domain and range, and apply transformations to the graphs of functions.

What this dot point is asking

A function is a rule that takes an input and produces exactly one output, and CCEA GCSE Further Mathematics uses function notation seriously. You must form composite functions (applying one after another), find inverse functions (the rule that undoes a function), state the domain and range, and transform graphs by translating, reflecting and stretching them. These skills tie together the algebra and the graph work of the whole unit.

Composite functions

A composite function chains two functions, and the order is crucial. The notation $fg(x)$ means substitute $g(x)$ into $f$, so $g$ acts first. Read it from right to left, like layers of brackets.

For example, with $f(x) = x + 3$ and $g(x) = 2x$, $fg(x) = f(2x) = 2x + 3$, whereas $gf(x) = g(x + 3) = 2(x + 3) = 2x + 6$. The two are different, which is exactly why the order matters in the exam.

Inverse functions

The inverse $f^{-1}$ reverses the action of $f$: if $f$ turns $3$ into $7$, then $f^{-1}$ turns $7$ back into $3$. To find it, set $y = f(x)$, rearrange to make $x$ the subject, then rewrite with $x$ as the input.

A useful check is that $ff^{-1}(x) = x$, because doing a function and then undoing it returns the original input. Graphically, the inverse is the reflection of the function in the line $y = x$, which swaps each $(a, b)$ for $(b, a)$.

Domain and range

The domain is the set of inputs a function is allowed to take, and the range is the set of outputs it produces. Some functions restrict the domain naturally: you cannot divide by zero, and you cannot take the square root of a negative number.

For $f(x) = \sqrt{x - 2}$, the domain is $x \ge 2$ so the root is defined, and the range is $f(x) \ge 0$ because a square root is never negative. Stating domain and range correctly is often worth marks in its own right, and a quick sketch makes the range easy to read.

Transforming graphs

Transformations move or reshape a graph in predictable ways. The rules split into changes outside the function (which affect $y$) and inside the function (which affect $x$, and act in the opposite sense to what you might expect).

Why this matters

Functions are the formal language for everything that follows: logarithms and exponentials are functions with their own inverses, trigonometric functions have ranges you must respect when solving equations, and calculus studies how a function changes. Graph transformations let you sketch new curves quickly from known ones, which saves time across the paper and supports the coordinate-geometry and trigonometry topics.