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

Recall our lesson on function composition.

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