To find \( f^{-1}(10) \), you need to determine the inverse of the function \( f(x) = -x^3 + 4 \) and then solve for \( x \) when the function equals 10.

Given:
\[ f(x) = -x^3 + 4 \]
We want to find \( x \) such that:
\[ f^{-1}(10) = x \text{ where } f(x) = 10 \]

Set the function equal to 10:
\[
-x^3 + 4 = 10
\]

Now, solve for \( x \):
\[
-x^3 = 10 - 4
\]
\[
-x^3 = 6
\]
\[
x^3 = -6
\]
\[
x = -\sqrt[3]{6}
\]

So, \( f^{-1}(10) = -\sqrt[3]{6} \).

### Do you want details or have any questions?

Here are 8 related questions:
1. How do you find the inverse of a cubic function in general?
2. Can the inverse of a function be found if the original function is not one-to-one?
3. What is the domain and range of the function \( f(x) = -x^3 + 4 \)?
4. How do you determine if a function is invertible?
5. What is the significance of the negative sign in front of \( x^3 \) in \( f(x) = -x^3 + 4 \)?
6. Can a function have more than one inverse?
7. What are the steps to graph \( f(x) = -x^3 + 4 \) and its inverse?
8. How do you interpret the result \( f^{-1}(10) = -\sqrt[3]{6} \) geometrically?

**Tip:** When solving for an inverse function, always ensure the original function is one-to-one (passes the horizontal line test), as only such functions have a proper inverse.

Discover how to find the inverse function f^-1(10) for the cubic function -x^3 + 4. Step-by-step solution provided includes determining the inverse and solving for x. Suitable for advanced high school students and anyone studying inverse functions.

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