Function notation

• Higher tier only.
• A function maps each element of one set (the domain) to exactly one element of another.
• Notation: f(x) = 3x − 1 or f : x ↦ 3x − 1. f(4) means the output when the input is 4.
• Some values must be excluded from the domain, e.g. x = 2 for f(x) = 1/(x − 2).
• fg(x) means do g first, then f. The inverse f⁻¹ undoes f.

One input, one output

A function is a mapping between two sets: every input from the first set is sent to exactly one output in the second. Both notations below mean the same thing.

Which inputs are allowed, which outputs are produced

The domain is the set of inputs; the range is the set of outputs. Exclude any input that would mean dividing by zero or taking the square root of a negative number.

fg means do g first, then f

fg(x) = f(g(x)): put x into g, then put the result into f. The function nearest the x acts first. In general fg(x) ≠ gf(x).

f⁻¹ undoes f

To find f⁻¹(x): write y = f(x), rearrange to make x the subject, then swap y for x. If x appears twice, use the "collect and factorise" method from changing the subject. Applying f then f⁻¹ returns the original input: ff⁻¹(x) = x.