College Algebra, Chapter 3, 3.6, Section 3.6, Problem 38

Determine the functions $f \circ g, \quad g \circ f, \quad f \circ f$ and $g \circ g$ and their domains if $f(x) = x^2$ and $g(x) = \sqrt{x-3}$
For $f \circ g$

$
\begin{equation}
\begin{aligned}
f \circ g &= f(g(x)) && \text{Definition of } f \circ g\\
\\
f \circ g &= \left( \sqrt{x-3} \right)^2 && \text{Definition of } g\\
\\
f \circ g &= x -3 && \text{Definition of } f
\end{aligned}
\end{equation}
$

The domain of the function is $(-\infty, \infty)$

For $g \circ f$

$
\begin{equation}
\begin{aligned}
g \circ f &= g(f(x)) && \text{Definition of } g \circ f\\
\\
g \circ f &= \sqrt{(x^2)-3} && \text{Definition of } f\\
\\
g \circ f &= \sqrt{x^2 - 3} && \text{Definition of } g
\end{aligned}
\end{equation}
$

Since the function involves square root we want

$
\begin{equation}
\begin{aligned}
x^2 -3 &\geq 0\\
\\
x^2 &\geq 3\\
\\
x &\geq \pm \sqrt{3}
\end{aligned}
\end{equation}
$

Thus, the domain is $[-\sqrt{3},\infty)$

For $f \circ f$

$
\begin{equation}
\begin{aligned}
f \circ f &= f(f(x)) && \text{Definition of } f \circ f\\
\\
f \circ f &= (x^2)^2&& \text{Definition of } f\\
\\
f \circ f &= x^4 && \text{Definition of } f
\end{aligned}
\end{equation}
$

The domain of the function is $(-\infty,\infty)$

For $g \circ g$

$
\begin{equation}
\begin{aligned}
g \circ g &= g(g(x)) && \text{Definition of } g \circ g\\
\\
g \circ g &= \sqrt{\sqrt{x-3}-3} && \text{Definition of } g\\
\end{aligned}
\end{equation}
$

Since the equation involves square root, we want

$
\begin{equation}
\begin{aligned}
\sqrt{x-3} - 3 &\geq 0 && \text{Add }3 \\
\\
\sqrt{x-3} &\geq 3 && \text{Square both sides}\\
\\
x- 3 &\geq 9 && \text{Add } 3\\
\\
x &\geq 12
\end{aligned}
\end{equation}
$

Thus, the domain of $g \circ g$ $[12,\infty)$