## Calculus (3rd Edition)

$(f\circ g)(t)=\sqrt{1-t^3}$; $D_{f\circ g}=(-\infty,1]$ $(g\circ f)(t)=1-t\sqrt t$; $D_{g\circ f}=[0,\infty)$
We are given the functions: $f(t)=\sqrt t$ $g(x)=1-t^3$ Determine the domains of $f$ and $g$: $D_f=(0,\infty)$ $D_g=(-\infty,\infty)$ Determine $(f\circ g)$: $(f\circ g)(t)=f(g(t))=f(1-t^3)=\sqrt{1-t^3}$ The function $(f\circ g)$ is defined for the values of $t$ for which $1-t^3\geq 0$: $1-t^3\geq 0$ $(1-t)(1+t+t^2)\geq 0$ $1-t\geq 0$ $t\leq 1$ $D_{f\circ g}=(-\infty,1]$ Determine $(g\circ f)$: $(g\circ f)(x)=g(f(x))=g\left(\sqrt t\right)=1-(\sqrt t)^3=1-t\sqrt t$ The domain of $(g\circ f)$ is: $D_{g\circ f}=[0,\infty)$