Domain of the definition of a composite function

Question

If $$f(x)= \sqrt{3|x|-x-2} \\ g(x)=\sin(x),$$ then the domain of definition of $(f\circ g)(x)$ is ...?

How do you calculate the domain of a composite function like this?

Answer 1

Hint:

Observe that $f$ is defined, as a real valued function, if and only if \begin{align*} &&3|x|-x-2&\ge 0\\ \iff && 3|x|&\ge x+2\\ \iff & &9x^2\ge x^2+4x+4 \quad&\text{or}\quad x+2\le 0\\ \iff&&2x^2-x-1\ge0\quad&\text{or}\quad x\le -2\\ \iff&&x\le-\frac{1}{2} \qquad\text{or}\qquad x\ge1\quad&\text{or}\quad x\le -2\\ \iff&&x&\in\big(-\infty,-\frac{1}{2}\big]\cup[1,\infty) \end{align*} Now, look for the values of $x$ such that $$\sin x \in\big(-\infty,-\frac{1}{2}\big]\cup[1,\infty)$$