College Algebra, Chapter 3, 3.6, Section 3.6, Problem 54

Given $\displaystyle H(x) = \sqrt{1+ \sqrt{x}} $, find functions $f$ and $g$ such that $F = f \circ g$
Since the formula for $H$ says to first take the square root then add 1 and then take the square root. We let
$g(x) = 1 + \sqrt{x}$ and $f(x) = \sqrt{x}$

$
\begin{equation}
\begin{aligned}
\text{Then }(f \circ g)(x) &= f(g(x)) && \text{Definition of } f \circ g\\
\\
(f \circ g)(x) &= f(1+\sqrt{x}) && \text{Definition of } g\\
\\
(f \circ g)(x) &= \sqrt{1+\sqrt{x}} && \text{Definition of } f\\
\\
(f \circ g)(x) &= H(x)
\end{aligned}
\end{equation}
$