Sheet № 29 · Higher only · AQA · Edexcel · OCR
Functions & Function Notation –
Functions and function notation are Higher-tier topics that underpin several areas of GCSE Maths — from graph transformations to inverse and composite functions. Understanding what f(x) means, how to evaluate a function at a given value, and how to set up and solve equations involving functions are all skills that AQA, Edexcel and OCR exp
§Key definitions
Question:
f(x) = 5x − 3. Find: (a) f(4), (b) f(−2), (c) the value of x when f(x) = 22.
(a)
f(4) = 5(4) − 3 = 20 − 3 = 17
(b)
f(−2) = 5(−2) − 3 = −10 − 3 = −13
(c)
Set 5x − 3 = 22:
Question 1:
f(x) = 7 − 2x. Find f(5) and f(−3).
§Formulas to memorise
f(x) = 2x + 5
x = (−b ± √(b² − 4ac)) / 2a
Input — the value you substitute into the function (the x-value).
Output — the value the function produces (the y-value or f(x) value).
Domain — the set of all possible input values.
Range — the set of all possible output values.
Write down the function — definition: f(x) = ...
Replace every x — in the expression with the given input value.
Calculate — using the correct order of operations (BIDMAS).
Set the function expression equal to k — e.g., 2x + 5 = 13.
Worked example
f(x) = 5x − 3. Find: (a) f(4), (b) f(−2), (c) the value of x when f(x) = 22.
(a) f(4) = 5(4) − 3 = 20 − 3 = 17
⚠ Common mistakes
- ✗Confusing f(x) with f × x. f(x) does not mean f multiplied by x. It means "the function f applied to x". Treat the brackets as instruction brackets, not multiplication brackets.
- ✗Squaring negatives incorrectly. When evaluating f(−3) for f(x) = x², remember (−3)² = 9, not −9. The brackets mean you square the whole number, including the negative.
- ✗Forgetting to substitute everywhere. If f(x) = 2x² + x, then f(3) = 2(3)² + (3) = 18 + 3 = 21. Do not forget the second x.
- ✗Not checking solutions. Always substitute your answer back into the original function to verify it gives the correct output.
✦ Exam tips
- →Write the substitution out in full. Show f(3) = 2(3)² + (3) before simplifying. This earns method marks even if you make an arithmetic error.
- →When asked "find x such that f(x) = k", set up and solve an equation. Do not try to work backwards informally — show clear algebraic working.
- →On AQA papers, function questions often appear alongside composite or inverse functions. Knowing basic evaluation fluently saves time for the harder parts.
- →If the function is quadratic, expect two solutions when solving f(x) = k. Do not stop after finding only one.