College Algebra, Chapter 3, Review Exercises, Section Review Exercises, Problem 80

If $\displaystyle T(x) = \frac{1}{\sqrt{1+\sqrt{x}}}$ find functions $f, g,$ and $h$ such that $f \circ g \circ h = T$.
$T = f(g(h(x)))$
The formula says that, first, take the square root and add 1. Then, take the square root and lastly the result is the divisor of 1. So we let
$h(x) = 1 + \sqrt{x}, \quad g(x) = \sqrt{x} \quad$ and $\quad \displaystyle f(x) = \frac{1}{x}$
Then, by checking

$
\begin{equation}
\begin{aligned}
f(g(h(x))) &= f(g(1+\sqrt{x}))\\
\\
&= f\left(\sqrt{1+\sqrt{x}}\right)\\
\\
&= \frac{1}{\sqrt{1+\sqrt{x}}}
\end{aligned}
\end{equation}
$