Determine $(g\circ f)(x)$ when $f(x)=-2\,x^2-4$ and $g(x)=-x$.

Recall our lesson on function composition. Our first step in determining $(g\circ f)(x)$ is to replace $x$ by $f$ in $g(x)$. This gives $g(f)=-f$. Next, place the formula $f(x)$ into every value of $f$ in $g(f)$ to obtain \begin{eqnarray}(g\circ f)(x)&=&g(f(x))\\&=&-(-2\,x^2-4)\\&=&2\,x^2+4\end{eqnarray}