Functions

A function is a rule that turns an input into exactly one output. Composite functions apply one rule, then feed the result into another.

$f(x) = 2x + 1 \qquad g(x) = 3x - 2$

Composite function fg(x) = f(g(x))

$g(4) = 3(4) - 2 = 10$

$f(10) = 2(10) + 1 = 21$

$fg(4) = 21$

Work from the inside out: apply g first, then apply f to that result. Check: gf(4) works the other way and gives $25$ (not the same, unless it happens to match).

Inverse function f⁻¹(x)

$y = 2x + 1$

Swap x and y, then rearrange to make y the subject:

$x = 2y + 1 \;\Rightarrow\; y = \frac{x - 1}{2}$

$f^{-1}(x) = \frac{x - 1}{2}$

Check: $f^{-1}(9) = 4 = 4$ -- feeding f(x) back through f⁻¹ returns the original input.