For each pair of functions $f$ and $g$ below, find $f(g(x))$ and $g(f(x))$.
Then, determine whether $f$ and $g$ are inverses of each other.
Simplify your answers as much as possible.
(Assume that your expressions are defined for all $x$ in the domain of the composition.
You do not have to indicate the domain.)
(a) $f(x)=x-2$
\[
g(x)=x+2
\]
\[
f(g(x))=
\]
\[
g(f(x))=
\]
$f$ and $g$ are inverses of each other
$f$ and $g$ are not inverses of each other
(b) $f(x)=-\frac{1}{6 x}, x \neq 0$
\[
g(x)=\frac{1}{6 x}, x \neq 0
\]
\[
f(g(x))=
\]
\[
g(f(x))=
\]
$f$ and $g$ are inverses of each other
$f$ and $g$ are not inverses of each other

Step 1 ：Given the functions $f(x) = x - 2$ and $g(x) = x + 2$, we need to find $f(g(x))$ and $g(f(x))$.

Step 2 ：First, find $f(g(x))$ by substituting $g(x)$ into $f(x)$: $f(g(x)) = (x + 2) - 2 = x$.

Step 3 ：Next, find $g(f(x))$ by substituting $f(x)$ into $g(x)$: $g(f(x)) = (x - 2) + 2 = x$.

Step 4 ：Since both $f(g(x))$ and $g(f(x))$ equal $x$, $f(x)$ and $g(x)$ are inverses of each other.

Step 5 ：Given the functions $f(x) = -\frac{1}{6x}$ and $g(x) = \frac{1}{6x}$, we need to find $f(g(x))$ and $g(f(x))$.

Step 6 ：First, find $f(g(x))$ by substituting $g(x)$ into $f(x)$: $f(g(x)) = -\frac{1}{6(\frac{1}{6x})} = -x$.

Step 7 ：Next, find $g(f(x))$ by substituting $f(x)$ into $g(x)$: $g(f(x)) = \frac{1}{6(-\frac{1}{6x})} = -x$.

Step 8 ：Since both $f(g(x))$ and $g(f(x))$ equal $-x$, which is not equal to $x$, $f(x)$ and $g(x)$ are not inverses of each other.

Step 9 ：\(\boxed{\text{(a) } f(x)=x-2 \text{ and } g(x)=x+2 \text{ are inverses of each other.}}\)

Step 10 ：\(\boxed{\text{(b) } f(x)=-\frac{1}{6 x}, x \neq 0 \text{ and } g(x)=\frac{1}{6 x}, x \neq 0 \text{ are not inverses of each other.}}\)