Let's say we have $f(x) = y = 1 - x^2,$ this is a straight forward composition if we compose $f$ with some other elementary function $g(x)$ to comprise $f(g(x)) = 1 - g(x)^2.$
But, let's say we are given $f(x,y) = z = 1 - x^2 -y^2.$
Well, what does it mean to compose $f(x,y)$ with another function $g(x,y)?$
Function $f$ maps $\mathbb R^2 \rightarrow \mathbb R^1$. You can only talk of $f(g(u,v))$ if $g$ maps $\mathbb R^2 \rightarrow \mathbb R^2$, which means that the map is represented by two functions $x = g_1(u,v)$ and $y=g_2(u,v)$. Then the composition is defined as $(f\circ g) (u,v) = f(g_1(u,v), g_2(u,v))$