The image contains a problem asking to determine whether two functions are inverses of each other. The functions provided are $f(x) = x^2 + 2$ and $g(x) = \sqrt{x - 2}$. The problem asks you to:

1. Find $f(g(x))$.
2. Find $g(f(x))$.
3. Determine if $g$ is the inverse of $f$ by evaluating $f(g(x))$ and $g(f(x))$.

Let's solve this step by step:

Step 1: Calculate $f(g(x))$:

Given $f(x) = x^2 + 2$ and $g(x) = \sqrt{x - 2}$, substitute $g(x)$ into $f(x)$:

$f(g(x)) = f(\sqrt{x - 2}) = (\sqrt{x - 2})^2 + 2$

Simplify:

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

Step 2: Calculate $g(f(x))$:

Now, substitute $f(x)$ into $g(x)$:

$g(f(x)) = g(x^2 + 2) = \sqrt{(x^2 + 2) - 2}$

Simplify:

$g(f(x)) = \sqrt{x^2} = |x|$

Step 3: Determine if $g$ is the inverse of $f$:

• For $f(g(x)) = x$, this holds true, which is the requirement for $g$ to be the inverse of $f$.
• For $g(f(x))$, we get $|x|$. Since $g(f(x))$ does not equal $x$ in all cases (it equals $x$ if $x \geq 0$ and $-x$ if $x < 0$), $g(x)$ is not the inverse of $f(x)$.

So, $g(x)$ is not the inverse of $f(x)$.

Do you want details or have any questions?

Here are some related questions for further exploration:

1. What are the general steps to verify if two functions are inverses of each other?
2. How does the domain and range of $f(x)$ and $g(x)$ affect their relationship as potential inverses?
3. Can a function have more than one inverse? Why or why not?
4. How does the property $g(f(x)) = |x|$ change the interpretation of $g(x)$ as an inverse function?
5. What modifications could be made to $f(x)$ or $g(x)$ to ensure they are inverses?

Tip: Always check both $f(g(x))$ and $g(f(x))$ to determine if two functions are truly inverses. The relationship must hold for all $x$ in the domain of the functions.