A-Level Maths Functions: Domain, Range, Composite and Inverse Functions
Functions become easier when you stop treating the notation as a collection of separate rules. A function is a mapping: it takes an allowed input, applies a defined process and produces an output. The domain tells you which inputs are allowed; the range tells you which outputs are actually produced; a composite function applies one function and then another; and an inverse function reverses a one-to-one mapping.
These ideas belong together. If a student can identify the input, follow the order of operations and check the restrictions, most function questions become a controlled sequence rather than a notation puzzle.
Exam-board specifications use functions within pure mathematics. Pearson Edexcel explicitly includes one-one and many-one mappings, domain and range, composite functions, inverse functions and the reflection of an inverse graph in the line y = x. OCR also places algebra and functions within the A-Level pure mathematics content. Students should still check the notation and scope used by their own course.
Begin with function notation
Suppose
f(x) = 2x − 3.
This does not mean that f is multiplied by x. It means: take the input x, multiply it by 2, then subtract 3. Therefore
f(5) = 2(5) − 3 = 7.
The safest habit is to substitute the entire input in brackets. If the input is x + 4, write
f(x + 4) = 2(x + 4) − 3 = 2x + 5.
That visible bracket prevents a common error: applying the function to only the first term. It also prepares students for compositions, where the input to one function is another algebraic expression.
A quick check is to describe the function in words before calculating. If the algebra and the verbal process disagree, the substitution needs revisiting.
Domain and range are part of the function
The domain is not an optional note added after the calculation. It determines where the function exists in the number system being used.
For
h(x) = 1 ÷ (x − 4),
the input x = 4 is not allowed because it makes the denominator zero. Over the real numbers, the domain is all real values except 4. The function can never produce 0, so 0 is also excluded from its range.
For
k(x) = √(x + 2),
real-valued outputs require x + 2 ≥ 0. The domain is therefore x ≥ −2. Because a principal square root is never negative, the range is k(x) ≥ 0.
Students often find a correct algebraic expression but lose the meaning of a result because they ignore these restrictions. Before manipulating a function, ask two questions:
1. Which inputs would make an operation undefined over the real numbers?
2. Given the shape and restrictions of the function, which outputs can it actually produce?
A sketch can help. It does not replace the algebra, but it can reveal a missing asymptote, endpoint or excluded value.
Composite functions: read the order from the inside
Let
f(x) = 2x − 3 and g(x) = x² + 1.
Using the convention in the Pearson Edexcel specification, fg(x) means f(g(x)): apply g first, then f.
Start with the inside function:
g(x) = x² + 1.
Now use that whole expression as the input to f:
f(g(x)) = 2(x² + 1) − 3 = 2x² − 1.
Reversing the order gives a different result:
g(f(x)) = (2x − 3)² + 1 = 4x² − 12x + 10.
This is why “do the inside first” is more reliable than trying to remember the order as a slogan. Write the nested notation before expanding. Then place the inner expression in brackets every time it replaces x.
Restrictions also travel through a composition. For f(g(x)) to exist, the input x must be allowed by g, and the output g(x) must lie in the domain of f. A final expression may look valid for every real x even when the original composite was not. Preserve any restrictions created earlier in the process.
Inverse functions reverse a one-to-one mapping
An inverse function undoes the original function. It is not the reciprocal 1 ÷ f(x).
Take
f(x) = 3x + 5.
Write y = 3x + 5, interchange x and y, then solve for y:
x = 3y + 5
y = (x − 5) ÷ 3.
Therefore
f⁻¹(x) = (x − 5) ÷ 3.
Check the result by composing the functions:
f⁻¹(f(x)) = ((3x + 5) − 5) ÷ 3 = x.
The domain and range swap roles. The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f. Graphically, the graphs of a function and its inverse are reflections in the line y = x.
An inverse function exists only when each output corresponds to one input on the chosen domain. The function q(x) = x² is not one-to-one across all real numbers because, for example, 2 and −2 both produce 4. Restricting the domain to x ≥ 0 makes it one-to-one, and the inverse is then q⁻¹(x) = √x, with the appropriate domain and range.
That restriction is not a technical afterthought. It is what makes the reverse mapping a function.
Diagnose the error before doing more practice
When a function question goes wrong, identify the first broken step. Common patterns include:
• reading fg(x) in the wrong order;
• substituting without brackets;
• confusing an inverse with a reciprocal;
• finding an algebraic inverse without checking whether the original function is one-to-one;
• ignoring a restricted domain after simplifying;
• stating a range that does not match the graph or the permitted inputs.
The correction should match the error. If order is the problem, practise writing nested notation before any algebra. If restrictions are the problem, annotate the domain beside every function before composing or inverting it. If the inverse process is unreliable, verify every answer by composition instead of completing more examples unchecked.
A practical functions revision routine
Use a short mixed set rather than a page of one repeated question type. For each function:
1. State the domain and identify any restrictions.
2. Calculate one value using clear substitution.
3. Form a composition in both orders and compare the results.
4. Decide whether the function is one-to-one on the stated domain.
5. If an inverse exists, find it and verify it by composition.
6. Sketch the original and inverse graphs where appropriate, including y = x.
Finish by explaining one answer aloud: what the input is, which operation happens first, why the domain is valid and how the check confirms the result. A student who can explain those decisions is less dependent on remembering a fragile procedure.
Parents supporting revision can ask, “Which inputs are allowed?” and “How do you know the inverse really undoes the function?” Those questions reveal understanding without turning the session into a second lesson.
Students who need to strengthen the algebra and graph skills underneath this topic can revisit CoPhil’s A-Level Maths bridge guide. CoPhil’s Subjects page outlines Mathematics support, while the Free Resources page provides further structured practice.
Functions are not four unrelated topics. They are one story about allowed inputs, resulting outputs and reversible or combined processes. Keep that story visible, and the notation becomes much easier to control.
Comments