https://file.aiquickdraw.com/mathbot/file/17258356304311itt8rzl.jpg

Let's solve the problem step by step.

### Given:
- \( f(x) = x^3 \)
- \( g(x) \) is defined as:
  \[
  g(x) = \begin{cases} 
  2x + 1 & \text{if } x \leq 0 \\
  x^2 - 1 & \text{if } x > 0 
  \end{cases}
  \]
- We need to find the value of \( g(f(f(x))) \) when \( x = -2 \).

### Step 1: Calculate \( f(x) \)
For \( x = -2 \),
\[
f(-2) = (-2)^3 = -8
\]

### Step 2: Calculate \( f(f(x)) = f(-8) \)
Since \( f(-2) = -8 \), we now need to find \( f(-8) \):
\[
f(-8) = (-8)^3 = -512
\]

### Step 3: Calculate \( g(f(f(x))) = g(-512) \)
Now we need to apply the value of \( -512 \) into \( g(x) \). Since \( -512 \leq 0 \), we use the first case of the piecewise function:
\[
g(-512) = 2(-512) + 1 = -1024 + 1 = -1023
\]

### Final Answer:
Since \( g(f(f(-2))) = -1023 \) is not one of the options provided, it seems there might be a miscalculation or a mistake in the problem as displayed. Let's double-check for any logical errors:

Actually, the options provided (like 0, -8, 9, etc.) don't match with -1023. This discrepancy suggests a potential issue in the problem setup or the given answer choices.

Do you want more details on a specific step, or perhaps to analyze a different part of this problem?

### Follow-up Questions:
1. What is the function of \( g(x) \) when \( x = 3 \)?
2. How would the value of \( g(f(f(x))) \) change if \( x = 1 \)?
3. What is the significance of piecewise functions in this problem?
4. What is the result of \( g(f(x)) \) if \( x = -1 \)?
5. How does the cubic function \( f(x) = x^3 \) behave for negative inputs?

**Tip:** Always double-check which piece of a piecewise function applies based on the input value when solving similar problems.

What is the value of g(f(x)) when x = -2 if f(x) = x^3 and g(x) = {2x + 1 when x ≤ 0, x^2 - 1 when x > 0}?

Learn how to evaluate the composite function g(f(x)) where f(x) = x^3 and g(x) is a piecewise function. Step-by-step guidance through understanding algebraic functions and solving problems involving piecewise functions. Ideal for high school students in Grades 9-12.

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