IGCSE Maths / Functions and function notation
What is a function
Edexcel International GCSE Maths (4MA1), chapter 24: Functions and function notation, specification point 3.2a. Free lesson.
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A function is a rule that gives every allowed input exactly one output. Knowing this definition helps you use function notation and decide whether a rule is a function.
What makes a rule a function?
A function maps inputs to outputs. The key test is about each input: it must give exactly one output, and the same input must give the same answer each time. Saying only that a function is a rule is not enough in an exam answer. Two different inputs may give the same output. That is allowed. One input giving two different outputs is not allowed.
How to read f(x)
The letter f is the name of the rule. The value in brackets is the input you put into it. For example, f(4) means use 4 as the input. It does not mean f multiplied by 4. For f(x) = 2x + 5, the machine doubles the input and then adds 5.
How to test whether a rule is a function
Try an input and check how many outputs it produces. With x → 2x + 5, putting in 9 gives 23, and only 23. This rule is a function. With x → ±√x, putting in 9 gives both 3 and −3, so that rule is not a function. The rule x → x² is a function: 3 gives 9, and −3 also gives 9. The definition checks from input to output. Two inputs sharing an output does not break the rule.
How to substitute a negative input
Keep the whole negative input inside brackets when substituting it. This makes clear that the negative number is being squared, so its square is positive.