Function Composition: f(g(x)) vs g(f(x))

Function composition: a pipeline

$f(g(x))$ means "apply $g$ first, then feed the result into $f$." Think of it as a pipeline: input → $g$ → output₁ → $f$ → final output.

Step-by-step: $f(x) = x^2$, $g(x) = x + 3$

$f(g(x))$: Replace $x$ in $f$ with $g(x)$: $f(x+3) = (x+3)^2 = x^2 + 6x + 9$.

$g(f(x))$: Replace $x$ in $g$ with $f(x)$: $g(x^2) = x^2 + 3$.

They're different! $f(g(2)) = 5^2 = 25$, but $g(f(2)) = 4 + 3 = 7$. Composition is not commutative.

The key rule

In $f(g(x))$: the inner function ($g$) gets applied first. Replace the $x$ in the outer function's formula with the entire inner function.

Common mistake

Confusing $f(g(x))$ with $f(x) \cdot g(x)$. Composition is substitution, not multiplication!

Try it in the visualization

Both compositions are graphed. Trace an input through the pipeline: see it enter $g$, then $f$. The resulting curves are visibly different, proving $f \circ g \neq g \circ f$. $f(g(x)) \neq g(f(x))$ in general. For $f(x)=x^2, g(x)=x+3$: $f(g(x))=(x+3)^2$ vs $g(f(x))=x^2+3$.