FUNCTIONS OBTAINED BY COMPOSITION

Let $D \subseteq ℝ^n$, where $n>1$ and let $f:D \rightarrow ℝ^m$ be a multivariable function, where $m⩾1$.

Let $Range(f) \subseteq E \subseteq ℝ^m$ and let $g:E \rightarrow ℝ^p$, where $p⩾1$.

For each $x\in D$ we will have $f(x)\in ℝ^n$. Since, $Range(f) \subseteq E$, we can say that $f(x)\in E$. Therefore, $g(f(x))$ is well defined and $g(f(x))\in ℝ^p$

$g \circ f:D \rightarrow ℝ^p$ is a multivariable function. It is called as the composition of $f$ and $g$. It is defined as follows:

$$g\circ f (x)=g(f(x))$$