Composition of Functions (gof and fog)

Reading the notation

$gof$ means "first apply $f$, then apply $g$ to the result", read right to left: $(g \circ f)(x) = g(f(x))$. Likewise $fog(x) = f(g(x))$. There's no reason these have to match.

Worked example

$f,g: \mathbb{R} \to \mathbb{R}$ are given by $g(x) = x^{1/3}$ and $f(x) = 8x^3$. Find $gof$ and $fog$.

Solution, gof: substitute $f(x)$ into $g$:

fog: substitute $g(x)$ into $f$:

So $gof(x) = 2x$ but $fog(x) = 8x$, a clean reminder that composition order matters.