How to find domain of complicated composite functions

Question

If I was asked to find the domain of arccos($e^x$), are there universal steps I can take to be able to find the domain? I know that you want the inner function in f(g(x)) to be defined as well as the function whole but I'm not sure how to think this one through...

Answer 1

$e^x$ is defined everywhere, but for $\arccos(t)$ you want $-1 \le t \le 1$. Now $e^x > 0 > -1$, but what do you need to get $e^x \le 1$?

Answer 2

For $f(g(x))$ to make sense, one needs $g(x)\in\mathrm{dom}(f)$, so $\mathrm{dom}(f\circ g) = g^{-1}(\mathrm{dom}(f))$. With $f=\arccos$ and $g=\exp$, you have $\mathrm{dom}(f)=[-1,1]$ and $\mathrm{dom}(f\circ g)=g^{-1}([-1,1])=(-\infty,0]$.