Let's analyze the functions given in the image.

We are provided with:

$f(x) = x + 2$ $g(x) = x - 2$

Part (a): $f(g(x))$

To compute $f(g(x))$, we substitute $g(x) = x - 2$ into the function $f(x) = x + 2$:

$f(g(x)) = f(x - 2) = (x - 2) + 2 = x$

Thus, $f(g(x)) = x$.

Part (b): $g(f(x))$

Similarly, for $g(f(x))$, substitute $f(x) = x + 2$ into the function $g(x) = x - 2$:

$g(f(x)) = g(x + 2) = (x + 2) - 2 = x$

Thus, $g(f(x)) = x$.

Part (c): Type of function

Since $f(g(x)) = x$ and $g(f(x)) = x$, this means that $g(x)$ is the inverse function of $f(x)$.

Summary of answers:

(a) $f(g(x)) = x$
(b) $g(f(x)) = x$
(c) Thus, $g(x)$ is called an inverse function of $f(x)$.

Would you like further details or have any questions?

Here are five related questions for practice:

1. How do you verify that two functions are inverses of each other?
2. What is the inverse of $f(x) = 2x + 3$?
3. If $f(x) = x^2$, is there an inverse function for all $x$?
4. What are the properties of inverse functions?
5. How do you graph a function and its inverse?

Tip: When finding the inverse of a function, swap $x$ and $y$ in the equation and solve for $y$.