https://file.aiquickdraw.com/mathbot/file/1723677014156tlinn5q5.jpg

The piecewise function given in the image is:

\[
f(x) = 
\begin{cases} 
x^3 - 2a & \text{if } x > -1 \\
6x^2 + 5a & \text{if } x \leq -1
\end{cases}
\]

We need to find \( f(a^2) \).

### Step 1: Determine which piece of the function applies
- If \( x = a^2 \) and \( a^2 > -1 \), the first piece \( f(x) = x^3 - 2a \) applies.
- Since \( a^2 \) is a non-negative number (because \( a^2 \geq 0 \)), it must be that \( a^2 > -1 \) for all real numbers \( a \).

Thus, \( f(a^2) = (a^2)^3 - 2a = a^6 - 2a \).

### Final Answer:
\[
f(a^2) = a^6 - 2a
\]

Would you like more details or have any questions?

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Here are 8 related questions you might find interesting:

1. What is \( f(a^2) \) when \( a = 0 \)?
2. How does the value of \( f(a^2) \) change as \( a \) increases?
3. Can \( f(a^2) \) be equal to zero? If so, for what values of \( a \)?
4. What would \( f(a^2) \) be if the function was defined differently for \( x \leq -1 \)?
5. How would you find the derivative of \( f(a^2) \) with respect to \( a \)?
6. What is the behavior of the function \( f(x) \) as \( x \) approaches infinity?
7. How would you integrate \( f(a^2) \) with respect to \( a \)?
8. What happens to \( f(a^2) \) as \( a \) approaches zero?

**Tip:** When working with piecewise functions, always check which condition applies based on the given input.

Learn how to find f(a^2) in a piecewise function problem with a detailed step-by-step solution. This problem involves evaluating a piecewise function for a specific input and covers mathematical concepts related to piecewise functions. Suitable for advanced high school students and anyone interested in mathematical problem-solving.

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