Let's solve the given problems step by step.

We have two functions:

• $f(x) = x + 6$
• $g(x) = x - 6$

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

To find $f(g(x))$, substitute $g(x)$ into $f(x)$:

$f(g(x)) = f(x - 6) = (x - 6) + 6 = x$

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

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

To find $g(f(x))$, substitute $f(x)$ into $g(x)$:

$g(f(x)) = g(x + 6) = (x + 6) - 6 = x$

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

(c) What kind of function is $g(x)$ relative to $f(x)$?

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

So, the answers are:

(a) $f(g(x)) = x$

(b) $g(f(x)) = x$

(c) $g(x)$ is called an inverse function of $f(x)$.

Would you like more details or have any questions about this solution?

Here are five related questions to expand your understanding:

1. How do you find the inverse of a function algebraically?
2. What are the conditions for a function to have an inverse?
3. Can every function have an inverse function? Why or why not?
4. How does the graph of a function relate to the graph of its inverse?
5. What is the significance of the composition $f(g(x)) = x$ in terms of function theory?

Tip: To verify that two functions are inverses, you must check both compositions $f(g(x))$ and $g(f(x))$ to see if they both return $x$.