We are given the function $g(x) = x^3 + e^x$ and the equation $g(x) = f^{-1}(x^2)$. Our task is to find the value of $f(0)$.

Let's analyze the information:

1. From $g(x) = f^{-1}(x^2)$, we understand that $f(g(x)) = x^2$, because applying $f$ to both sides of $g(x) = f^{-1}(x^2)$ yields $f(g(x)) = x^2$.

2. To find $f(0)$, we need to determine the value of $g(a)$ such that $f(g(a)) = 0$. This would imply $a = 0$, so we evaluate $g(0)$: $g(0) = 0^3 + e^0 = 0 + 1 = 1.$

3. Therefore, $f(1) = 0$.

Answer: The correct option is 1.

Would you like further details on this solution, or do you have any questions?

Here are some related questions for further practice:

1. How do you find the inverse of a function like $g(x)$?
2. If $g(x) = x^3 + e^x$, what is $g'(x)$?
3. How does the composition of functions affect the process of solving inverse-related equations?
4. What if $g(x) = x^3 + 2e^x$; how would that affect $f(0)$?
5. How would the answer change if $g(x) = x^3 - e^x$?

Tip: Remember that to find the inverse function, you need to ensure that each element in the domain maps uniquely to an element in the range.