Single Variable Calculus, Chapter 1, 1.3, Section 1.3, Problem 49

Find an expression of the function in the form $f \circ g \circ h$

Given: $H(x) = \sec ^4 (\sqrt{x}) $

The function $H(x) = \sec ^4 (\sqrt{x})$ states that we first take the square root then we take the secant of the result and raise it to $4^{th}$ power. So we have,

$\fbox{$ \displaystyle h(x) = \sqrt{x} \qquad g(x) = \sec x \qquad f(x) =x^4$}$

Upon checking:


$
\begin{equation}
\begin{aligned}

f \circ g \circ h = & f(g(h(x)))\\
f(g(\sqrt{x})) = & \sec x\\
f(\sec \sqrt{x}) = & x^4\\
f \circ g \circ h = & \sec^4 \sqrt{x}

\end{aligned}
\end{equation}
$