# Math Help - Range of f(x)?

1. ## Range of f(x)?

Given the functions
$f(x) = \exp(\arccos \left(\sin \left(x + \frac{\pi}{3}\right)\right))$,
$g(x) = \mbox{arccsc} \left(\frac{4 - 2\cos x}{3}\right)$
and the function $h(x) = f(x)$ defined only for the common domain of $f(x)\ \mbox{\&}\ g(x)$.

Calculate range of $h(x)$.

2. Since sines and cosines take on the same values,

$f(x)=\exp\left(\arccos\left(\sin\left(x+\frac{\pi} {3}\right)\right)\right)$

is defined everywhere, i.e., its domain is $\mathbb{R}$. The domain of $h$ is therefore just the domain of $g$, consisting of precisely the values of $x$ for which

$\frac{4-2\cos x}{3}$

belongs to $(-\infty,-1]\cup[1,\infty)$. When we find these values, we can use them to find the range of $h$ on that domain.

3. How do we find these values?