Functions

A function is a rule that turns each input into exactly one output. We write f(x). Composite functions do one function then another; an inverse function undoes the original.

Key facts

  • f(3) means put 3 into the rule.
  • fg(x) means do g first, then f.
  • To find f⁻¹(x): write y = f(x), swap x and y, rearrange for y.
  • Domain: the inputs allowed. Range: the outputs you get.

Worked examples

Example 1

f(x) = 2x + 1 and g(x) = x². Find fg(3) and gf(3).

  1. g(3) = 9, then f(9) = 19.
  2. f(3) = 7, then g(7) = 49.
  3. The order matters!

Answer: fg(3) = 19, gf(3) = 49

Example 2

Find the inverse of f(x) = (x − 4)/3.

  1. y = (x − 4)/3; swap: x = (y − 4)/3.
  2. 3x = y − 4 y = 3x + 4

Answer: f⁻¹(x) = 3x + 4