College Algebra, Chapter 3, 3.6, Section 3.6, Problem 40

Determine the functions $f \circ g, \quad g \circ f, \quad f \circ f$ and $g \circ g$ and their domains if $f(x) = x- 4$ and $g(x) = |x+4|$
For $f \circ g$,

$
\begin{equation}
\begin{aligned}
f \circ g &= f(g(x))\\
\\
f \circ g &= |x+4| -4
\end{aligned}
\end{equation}
$

By using the property of absolute value

$
f \circ g = |x +4| -4 \quad \Longrightarrow \quad f \circ g =
\begin{array}{c}
x + 4 - 4 & \text{for} & x > - 4\\
\\
-(x+4)-4 & \text{for} & x < -4
\end{array}
$


$
\phantom{f \circ g = |x +4| -4 \quad \Longrightarrow \quad } f \circ g =
\begin{array}{c}
x & \text{for} & x > - 4\\
\\
-x-8 & \text{for} & x < -4
\end{array}\\
$

Thus, the domain is $(-\infty,\infty)$

For $g \circ f$

$
\begin{equation}
\begin{aligned}
g \circ f &= g(f(x)) && \text{Definition } g\circ f\\
\\
g \circ f &= |x -4 + 4| && \text{Definition of } f\\
\\
g \circ f &= |x| && \text{Definition of } g
\end{aligned}
\end{equation}
$

Recall that
$
|x| = \left\{
\begin{array}{c}
x & x \geq 0\\
\\
-x & x < 0
\end{array}\right.
$
so the domain of $g \circ f$ is $(-\infty, \infty)$


For $f \circ f$

$
\begin{equation}
\begin{aligned}
f \circ f &= f(f(x)) && \text{Definition of } f \circ f\\
\\
f \circ f &= x -4 - 4 && \text{Definition of } f \\
\\
f \circ f &= x - 8 && \text{Defintion on } f
\end{aligned}
\end{equation}
$


The domain of $f \circ f$ 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 &= ||x + 4| + 4| && \text{Definition of } g
\end{aligned}
\end{equation}
$

By using property of absolute value

$
g \circ g = ||x +4| + 4| \quad \Longrightarrow \quad g \circ g =
\begin{array}{c}
x + 4 - 4 & \text{for} & x > - 4\\
\\
-(x+4)-4 & \text{for} & x < -4
\end{array}
$

Thus, the domain of $g \circ g$ is $(-\infty, \infty)$