The given transformation starts with the function:

\[
g(x) = \frac{1}{2} f(x) - 2
\]

and results in the following form:

\[
g(x) = \frac{1}{2}x^3 - 2
\]

This means that the function \( f(x) \) must be \( f(x) = x^3 \), since the transformation implies that \( f(x) \) has been replaced with \( x^3 \) in the final equation.

### Explanation:
1. The original form is \( g(x) = \frac{1}{2} f(x) - 2 \).
2. To match the final form, you substitute \( f(x) = x^3 \).
3. After substitution, the equation becomes \( g(x) = \frac{1}{2} x^3 - 2 \), which matches the given final form.

Would you like more details on how this transformation works?

Here are 5 related questions you can explore:
1. How do transformations affect the graph of a function?
2. What would happen if \( f(x) = x^4 \) instead of \( x^3 \)?
3. How can you solve for \( x \) if \( g(x) = 0 \)?
4. What is the derivative of \( g(x) \)?
5. How would adding a horizontal shift to \( f(x) \) change \( g(x) \)?

**Tip:** Always check the impact of each transformation step on the graph or function properties, like vertical stretching and translations!

g(x) = 1/2 f(x) - 2 -> g(x) = 1/2(x)^3 - 2

Explore how to transform the function g(x) = 1/2 f(x) - 2 into g(x) = 1/2 x^3 - 2. Understand the concept of function substitution and transformations. This problem is suitable for students in Grades 10-12, covering cubic functions and algebraic manipulations.

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