Answer
$f \circ g (x) = |sin(x)|$
$g \circ f (x) =sin(|x|)$.
Domains: $\mathbb{R}$
Work Step by Step
$f \circ g$ is defined by $f \circ g (x) = f(g(x)) = |sin(x)|$ and
$g \circ f$ is defined by $g \circ f (x) = g(f(x)) = sin(|x|)$.
Note: the domain of $f$ is $\mathbb{R}$ (all real numbers) and the domain of $g$ is $\mathbb{R}$. Hence, we have:
The domain of $f \circ g$ is given by
$D_{f \circ g} = \{x \in D_g : \, g(x) \in D_f\} = \{x \in \mathbb{R} : \, |x| \in \mathbb{R}\}$
and then the domain of $f \circ g$ is equal to $\mathbb{R}$.
The domain of $g \circ f$ is given by
$D_{g \circ f} = \{x \in D_f : \, f(x) \in D_g\} = \{x \in \mathbb{R} : \, sin(x) \in \mathbb{R}\}$
and then the domain of $g \circ f$ is equal to $\mathbb{R}$.
