Understand and use composite functions; inverse functions and their graphs

Composite and inverse functions: Edexcel A-level Mathematics

Overview

Specification point 2.8 asks you to treat functions as mappings, combine functions in the correct order, and reverse a function when that reversal is valid. This is not merely notation. In an Edexcel question, a composite function can expose whether you understand an algebraic process as a chain of actions, while an inverse-function question tests algebra, restrictions, and graph interpretation together. Expect short items such as finding fg(3) or f^{-1}(x), but also connected questions in which a domain restriction, a composite calculation and a sketch each earn marks. The topic links directly to transformations, solving equations, logarithms, exponentials, differentiation and modelling. In the full A-level, each paper is 2 hours and 100 marks, so use approximately 1.2 minutes per mark as your pacing guide.

Key concepts

1. A function is a rule with a controlled output

A function takes each allowed input and gives it exactly one output. The allowed inputs form the domain; the set of outputs actually produced is the range. Write f: x ↦ 2x + 5 to show that the rule sends x to 2x + 5, and write f(3) = 11 for a particular input-output calculation.

The distinction between one-to-one and many-one matters. A one-to-one function gives different outputs for different inputs. Its graph passes the horizontal-line test: no horizontal line meets it more than once. A many-one function can send different inputs to the same output. For example, x² is many-one on all real numbers because 2 and −2 both give 4.

Why does this matter? An inverse must take an output back to one unique original input. If two starting inputs share an output, an inverse would have to give two answers for one input, so it would not be a function. Restricting the domain can repair this. For f(x) = x², state x ≥ 0; then every output comes from one input and the inverse is f^{-1}(x) = √x.

Examiner focus: candidates must state a restriction clearly, for example x ≥ 0. Writing only √x after starting with x² loses credit if the question has not already supplied a suitable domain.

2. Composite functions: the right-hand rule comes first

For a composite function fg(x), do g first, then f. Read it from the inside out: calculate g(x), then feed that entire result into f. This mirrors function notation such as f(g(x)).

Suppose f(x) = 3x − 1 and g(x) = x² + 2.

fg(x) = f(g(x)) = f(x² + 2)

= 3(x² + 2) − 1 = 3x² + 5.

The reverse order is a different function:

gf(x) = g(f(x)) = g(3x − 1)

= (3x − 1)² + 2 = 9x² − 6x + 3.

The functions are not usually commutative, so never assume fg = gf. A reliable check is to write a small arrow above your working: x → g(x) → f(g(x)). It prevents the common error of applying the left-hand function first.

Why this works: the output of g becomes the input of f. Parentheses preserve the whole output, especially when the second rule involves squaring, a reciprocal, a logarithm or a modulus. If f(x) = x² and g(x) = 2x − 3, then f(g(x)) is (2x − 3)², not 4x² − 9. The bracket is a mathematical instruction, not decoration.

3. Domains and ranges travel through a composite function

When finding fg, the domain contains values that are allowed into g and whose resulting values are allowed into f. This becomes important with square roots, fractions and logarithms. If f(x) = √x and g(x) = 5 − x, then

fg(x) = √(5 − x).

For a real-valued answer, 5 − x ≥ 0, so the domain is x ≤ 5. A candidate who gives the formula but ignores the restriction may miss a final accuracy or independent mark in a domain question.

For rational functions, exclude inputs that make a denominator zero. For a logarithm, the input must be strictly positive. For a square root in the real-number setting, the expression inside must be non-negative. Sketching before you state the range is often faster and safer than relying on a remembered pattern.

4. Finding an inverse algebraically

Use a controlled swap-and-solve method:

1. Write y = f(x).
2. Interchange x and y.
3. Rearrange to make y the subject.
4. Rename the result f^{-1}(x).
5. State any necessary domain or range restriction.

For f(x) = (2x − 3)/5, begin with y = (2x − 3)/5. Swap: x = (2y − 3)/5. Then 5x = 2y − 3, so 2y = 5x + 3 and y = (5x + 3)/2. Therefore f^{-1}(x) = (5x + 3)/2.

Do not confuse f^{-1}(x) with 1/f(x). The inverse reverses the mapping; the reciprocal replaces a value by its multiplicative reciprocal. For instance, if f(x) = 2x + 1, then f^{-1}(x) = (x − 1)/2, whereas 1/f(x) = 1/(2x + 1). They are entirely different functions.

5. Verification: compose back to the identity

The defining test is:

f^{-1}(f(x)) = x and f(f^{-1}(x)) = x.

For the previous pair, substitute f(x) = (2x − 3)/5 into the inverse:

f^{-1}(f(x)) = [5((2x − 3)/5) + 3]/2 = (2x − 3 + 3)/2 = x.

This check is valuable when a rearrangement contains a negative sign, a fraction or a square root. In a longer response it can secure assurance before you commit to a final answer, but only show it when the command word is ‘show that’, ‘verify’ or the available marks justify it.

6. Inverse graphs: reflect, do not translate

The graph of y = f^{-1}(x) is the reflection of y = f(x) in the line y = x. Every point (a, b) on f becomes (b, a) on the inverse. Intercepts swap when they exist: an x-intercept (p, 0) becomes the y-intercept (0, p). The domain and range also swap.

Draw the line y = x as a dashed construction line. Then reflect recognisable points and shape, rather than trying to calculate many new coordinates. If the original graph is a restricted parabola, its inverse is a square-root-shaped curve. If the original graph is a non-horizontal straight line, its inverse is another straight line, with coordinates swapped. A vertical line cannot be a function; after reflection it would become a horizontal line. This explains why the original must be one-to-one before its inverse can be a function.

7. Relationships and formulae to memorise

All function relationships in this topic are must memorise. They are not supplied on the Mathematical Formulae and Statistical Tables booklet.

• fg(x) = f(g(x)): apply g, then apply f.
• gf(x) = g(f(x)): apply f, then apply g.
• f^{-1}(f(x)) = x: inverse after function returns the starting input, where defined.
• f(f^{-1}(x)) = x: function after inverse returns the starting input, where defined.
• (a, b) → (b, a): coordinate rule for reflection in y = x.
• domain(f) = range(f^{-1}) and range(f) = domain(f^{-1}).

Graph and data skills

On a sketch, label axes, any given domain endpoints, key intercepts and the line y = x when an inverse is requested. Use open circles for excluded endpoints and solid circles for included endpoints. For an inverse, swap at least two labelled coordinates to demonstrate a genuine reflection. Never draw the full sideways parabola for y = x² as an inverse function over all real numbers, because it fails the vertical-line test. A-Level Mathematics has no Foundation/Higher tier split, and there are no relevant unit conversions or required practicals for this specification point.

Mark-awareness: turning working into credit

Edexcel mark schemes commonly distinguish M marks for a valid method, A marks for accurate completion after the method, and B marks for an independently correct fact. In functions questions, an M mark may come from writing a correct composition such as f(g(x)), swapping x and y, or using a stated domain restriction. An A mark then depends on accurate expansion, rearrangement or final notation. Do not jump from a question to an unexplained answer in a multi-mark part. A clear line for the substitution or swap makes a creditable method visible, and it can protect follow-through credit if one later arithmetic slip occurs.

Practical applications and synoptic thinking

Composite functions model processes that occur in stages. A temperature sensor may convert temperature to voltage, then voltage to a displayed reading. Tax calculations, currency conversion with commission, and coordinate changes all behave like chained rules. An inverse function reverses a reversible process: a calibration equation can be undone to infer the original quantity from a display value.

In later work, composite functions sit behind the chain rule. When differentiating f(g(x)), first identify the inner function and the outer function, exactly as you do here. In exponentials and logarithms, ln(x) and e^x are inverse functions on their valid domains. In transformations, the distinction between changing the input and changing the output reinforces why function order matters. These links turn a short algebra topic into a recurring way of thinking across the whole course.

Examiner-ready checklist

Before moving on, ask: Have I applied the right-hand function first? Have I kept brackets around a substituted expression? Is the proposed inverse one-to-one on the stated domain? Have I swapped domain and range? If I have sketched an inverse, have I reflected in y = x, rather than shifted or rotated the graph? Those checks routinely protect marks in questions that look routine but combine several ideas.