What you'll learn
Functions are rules that turn an input into an output, and composite and inverse functions combine or reverse these rules. In this guide you will learn function notation, how to evaluate functions, how to form composite functions, how to find inverse functions, and how to use them correctly. These ideas appear in higher-tier GCSE algebra questions.
Key terms and definitions
Function — a rule that assigns exactly one output to each input, written f(x).
Function notation — f(x) means "the function f applied to x".
Composite function — applying one function to the result of another, e.g. fg(x).
Inverse function — the function that reverses f, written f⁻¹(x).
Input / output — the value put into a function and the value it produces.
Core concepts
Function notation and evaluating
f(x) describes a rule. To evaluate a function, substitute the input value. For f(x) = 2x + 3, f(5) = 2(5) + 3 = 13. You can also substitute expressions: f(a) = 2a + 3. Reading function notation correctly is the first step.
Composite functions
A composite function applies one function to the output of another. fg(x) means "do g first, then f": substitute g(x) into f. For example, if f(x) = 2x and g(x) = x + 1, then fg(x) = f(g(x)) = f(x + 1) = 2(x + 1) = 2x + 2. Note that fg(x) and gf(x) are usually different — order matters.
Working composites with numbers
To find fg(3), either substitute 3 into the composite formula, or work step by step: find g(3) first, then apply f to the result. Both give the same answer; the step-by-step approach reduces errors.
Inverse functions
The inverse function f⁻¹(x) reverses f: it takes an output back to its input. To find it:
- Write y = f(x).
- Rearrange to make x the subject.
- Swap x and y (or replace y with x) to write f⁻¹(x).
For f(x) = 2x + 3: y = 2x + 3 → x = (y − 3)/2 → f⁻¹(x) = (x − 3)/2.
Checking inverses
A correct inverse satisfies f(f⁻¹(x)) = x. You can check by substituting: applying f and then f⁻¹ (in either order) should return the original value.
Worked examples
Example 1: Evaluating
Given f(x) = 3x − 4, find f(6).
f(6) = 3(6) − 4 = 18 − 4 = 14.
Example 2: Composite
Given f(x) = x² and g(x) = x + 2, find fg(3).
g(3) = 3 + 2 = 5, then f(5) = 5² = 25.
Example 3: Inverse
Find the inverse of f(x) = 4x − 1.
y = 4x − 1 → x = (y + 1)/4 → f⁻¹(x) = (x + 1)/4.
Common mistakes and how to avoid them
Doing composites in the wrong order. fg(x) means apply g first, then f.
Assuming fg = gf. They are usually different — order matters.
Confusing f⁻¹ with 1/f. The inverse reverses the function; it is not the reciprocal.
Rearranging errors when finding inverses. Make x the subject carefully, step by step.
Substituting incorrectly. Use brackets when substituting expressions into functions.
Exam technique for Composite and Inverse Functions
Read the notation carefully — fg(x) is g then f.
Work composites step by step with numbers to reduce errors.
Find inverses by writing y = f(x), rearranging for x, and swapping variables.
Use brackets when substituting expressions.
Check inverses with f(f⁻¹(x)) = x.
Quick revision summary
A function f(x) is a rule giving one output per input; evaluate it by substituting (f(5) for f(x) = 2x + 3 gives 13). A composite function fg(x) means "do g first, then f" — substitute g(x) into f (f(x) = 2x, g(x) = x + 1 → fg(x) = 2x + 2); the order matters, so fg(x) ≠ gf(x) in general. With numbers, work composites step by step: find the inner function's value, then apply the outer. The inverse function f⁻¹(x) reverses f: write y = f(x), rearrange to make x the subject, then swap variables (f(x) = 2x + 3 → f⁻¹(x) = (x − 3)/2). Note f⁻¹ is the reverse process, not the reciprocal 1/f. Check an inverse with f(f⁻¹(x)) = x. The common errors are doing composites in the wrong order, assuming fg = gf, confusing inverse with reciprocal, and rearranging mistakes. Read notation carefully, work step by step, rearrange precisely, and verify your inverse.