The problem in the image can be translated as follows:

"If $g(x) = x^2 + 4$ with $x \geq 0$, then the value of $g^{-1}(13)$ is..."

Solution:

1. The function $g(x) = x^2 + 4$ is given, and we are asked to find $g^{-1}(13)$.

2. The inverse function, $g^{-1}(y)$, will return the value of $x$ such that $g(x) = y$.

3. Set $g(x) = 13$: $x^2 + 4 = 13$ Subtract 4 from both sides: $x^2 = 9$ Taking the square root of both sides, we get: $x = \sqrt{9} = 3$

   Since the condition $x \geq 0$ is given, we discard the negative root.

Therefore, the value of $g^{-1}(13)$ is $\boxed{3}$.

Would you like further explanation or have any other questions?

Here are five related questions to explore this concept further:

1. How do you determine the domain of an inverse function?
2. What is the significance of the condition $x \geq 0$ in this problem?
3. How would you graph both $g(x)$ and $g^{-1}(x)$ on the same plane?
4. What would change if the function were $g(x) = x^2 + 1$ instead of $x^2 + 4$?
5. How do you find the inverse of a general quadratic function?

Tip: When finding the inverse of a function, always make sure that the function is one-to-one (or restricted) over the given domain, so that an inverse exists.