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) $f(x)=-\frac{1}{6x},x\ne 0$
$$g(x)=\frac{1}{6x},x\ne 0$$
$$f(g(x))=$$
$$g(f(x))=$$
$f$ 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 ：$\boxed{{\displaystyle \text{(a) }f(x)=x-2\text{ and }g(x)=x+2\text{ are inverses of each other.}}}$

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