# Finding range of $f(g(h(x)))$

\begin{align*} f(x) &= \frac{2}{x+1}, \\ g(x) &= \cos x, \\ h(x) &= \sqrt{x+3} \end{align*} Find the range of $f(g(h(x)))$.

You want to all the possible "outputs" of the composition of the three functions. First note that the range of the inner most function $h$ is all non-negative numbers. Hence the "input" for the function $g$ is all non-negative real numbers. But for those you in $g(x) = \cos(x)$ get all real numbers between $-1$ and $1$ (both included). Note now that the domain of $f$ is all the real numbers that are not equal to $-1$, hence the possible "inputs" of $f$ is the interval $(-1, 1]$.
So now you just need to determine the possible values of $f$ when $x$ is in $(-1, 1]$. Hint: For this note that $f$ is a decreasing function.