Composite functions

#### Tier: #Higher

Description

A composite function applies one function to the result of another. The notation $fg(x)$ means "apply $g$ first, then $f$".

$$fg(x) = f(g(x))$$

Example: $f(x) = 2x + 1$ and $g(x) = x^2$. $$fg(x) = f(g(x)) = f(x^2) = 2x^2 + 1$$ $$gf(x) = g(f(x)) = g(2x+1) = (2x+1)^2$$

Note that in general $fg(x) \neq gf(x)$ — order matters.

Evaluating at a specific value: $$fg(3) = f(g(3)) = f(9) = 2(9)+1 = 19$$

Or substitute $x = 3$ directly into $fg(x) = 2x^2 + 1$: $2(9)+1 = 19$ ✓

Caution with notation: $fg(x)$ means $f$ applied after $g$ — the rightmost function is applied first. Some students misread this.

Common error: applying $f$ before $g$ in $fg(x)$. Always read right-to-left: $f$ of $g$ of $x$.

Links

Function notation Inverse functions Substitution