To solve the problem in the image, let's break it down step by step:

Given:

• $f(x) = x + 3$
• $g(x) = x - 3$

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

To find $f(g(x))$, we substitute $g(x)$ into the function $f(x)$.

1. $g(x) = x - 3$, so replace $x$ in $f(x) = x + 3$ with $g(x) = x - 3$: $f(g(x)) = (x - 3) + 3 = x$ Thus, $f(g(x)) = x$.

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

To find $g(f(x))$, we substitute $f(x)$ into the function $g(x)$.

1. $f(x) = x + 3$, so replace $x$ in $g(x) = x - 3$ with $f(x) = x + 3$: $g(f(x)) = (x + 3) - 3 = x$ Thus, $g(f(x)) = x$.

(c) What is $g(x)$ called?

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

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

Let me know if you want more details on any of the steps!

Here are five related questions to deepen your understanding:

1. What conditions must be met for two functions to be inverses of each other?
2. How can you verify that a function is invertible?
3. What is the graphical relationship between a function and its inverse?
4. Can a function have more than one inverse?
5. How do you find the inverse of more complex functions?

Tip: To confirm that two functions are inverses, check if both $f(g(x)) = x$ and $g(f(x)) = x$ hold true!