IGCSE Maths / Functions and function notation
Function notation and evaluation
Edexcel International GCSE Maths (4MA1), chapter 24: Functions and function notation, specification point 3.2b. Free lesson.
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Function notation gives a rule for turning an input into an output. You need to recognise both ways it is written and substitute inputs carefully, especially when they are negative or when combining functions.
How to read function notation
A function is a rule that changes an input into an output. The two common notations look different, but they mean the same thing. f(x) = 3x − 4 and f: x → 3x − 4 both say that the function named f takes an input x and applies the rule 3x − 4. For example, putting in 5 gives 3 × 5 − 4 = 11, so the output is 11 in either notation.
How to follow a function rule
Read the rule in the order of the operations. For f(x) = 3x − 4, multiply the input by 3 first, then subtract 4. Changing that order changes the function. For f(5), calculate 3 × 5 − 4 = 11. Subtracting first would give 3 × (5 − 4) = 3. That is not the rule 3x − 4.
How to substitute an input
To evaluate a function, replace every x in its rule with the input. If the input is negative, put it in brackets so the negative sign is included wherever the input is squared or multiplied. For f(x) = x² − 4x and g(x) = 5 − 2x, each function has its own rule. For f(−3), substitute −3 everywhere x appears: (−3)² − 4(−3) = 9 + 12 = 21. For g(4), substitute 4: 5 − 2 × 4 = −3. Function values can be negative.
How to combine two function values
When a question asks for f(2) + g(2), work out each function separately using the same input, then add the answers. For f(x) = x² − 4x, f(2) = 4 − 8 = −4. For g(x) = 5 − 2x, g(2) = 5 − 4 = 1. So f(2) + g(2) = −4 + 1 = −3. The fact that this matches g(4) is a coincidence, not a rule.
Why the input stays together
The bracket in f(1 + 2) marks the whole input. Add inside the bracket first, then apply the function once. You cannot split the input across two separate function values. For f(x) = 3x − 4, f(1) = −1 and f(2) = 2, so f(1) + f(2) = 1. But f(1 + 2) = f(3) = 3 × 3 − 4 = 5. Therefore f(1 + 2) ≠ f(1) + f(2). For g: x → 2x² + 3x, g(−2) = 2(−2)² + 3(−2) = 8 − 6 = 2.