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 $(-\infty, \infty)$