Functions

• Use function notation, such as f(x) = 3x − 5, and understand domain and range.
• Find inverse functions f⁻¹(x).
• Form composite functions: gf(x) = g(f(x)) means do f first, then g.
• Use mapping diagrams. Domains and ranges of composite functions are not required.

f(4) means substitute x = 4

f(x) = 3x − 5 is a rule. f(4) means replace x by 4: f(4) = 7. The domain is the set of input values; the range is the set of output values. A mapping diagram shows arrows from inputs to outputs.

Swap, then rearrange

The inverse function f⁻¹ undoes f. Write y = f(x), rearrange to make x the subject, then swap the letters to write f⁻¹(x).

gf(x) = g(f(x)): the function nearest x acts first

To find fg(x), put the whole of g(x) into f in place of x. Order matters: fg(x) and gf(x) are usually different. For a number, work from the inside out: fg(−1) means find g(−1), then put the answer into f.