Function notation; composite and inverse functions

Key learnings

• •

  f(x) means "substitute the value for x"

  $f(4)$ means replace every $x$ with 4 and evaluate. Solving $f(x) = 10$ means set the expression equal to 10 and solve for $x$.

• •

  Composite functions: inside first

  $fg(x)$ (f of g of x — apply g first, then f) means $f(g(x))$ — do $g$ first, then feed the result into $f$. The order matters: $fg(x)$ is usually not the same as $gf(x)$.

• •

  Inverse: swap and rearrange

  To find $f^{-1}(x)$ (the inverse of f), write $y = f(x)$, swap the roles of $x$ and $y$, then make $y$ the subject. The inverse undoes the function.