Answer
$f \circ g (x) = cos(x^3 +x^2)$
$g \circ f (x) = cos^3(x) + cos^2(x)$
Domains: $ \mathbb{R}$
Work Step by Step
$f \circ g$ is defined by $f \circ g (x) = f(g(x)) = cos(x^3 +x^2)$ and
$g \circ f$ is defined by $g \circ f (x) = g(f(x)) = cos^3(x) + cos^2(x)$.
The domain of $f$ is $\mathbb{R}$ and the domain of $g$ is $\mathbb{R}$. Hence, we have:
Domain of $f \circ g$ is given by
$D_{f \circ g} = \{x \in D_g : \, g(x) \in D_f\} = \{x \in \mathbb{R} : \, x^3+x^2 \in \mathbb{R}\}$
and then the domain of $f \circ g$ is equal to $\mathbb{R}$.
Domain of $g \circ f$ is given by
$D_{g \circ f} = \{x \in D_f : \, f(x) \in D_g\} = \{x \in \mathbb{R} : \, cos(x) \in \mathbb{R}\}$
and then the domain of $g \circ f$ is equal to $\mathbb{R}$.
