We are given the function \( f(x) = kx^3 \) and the information that \( f^{-1}(15) = 2 \).

This means that \( f(2) = 15 \), since \( f^{-1}(y) = x \) implies that \( f(x) = y \).

Now, use the given function to find \( f(2) \):
\[
f(2) = k(2)^3 = k \times 8 = 8k
\]

Since we know \( f(2) = 15 \), we can set up the equation:
\[
8k = 15
\]

Solving for \( k \):
\[
k = \frac{15}{8}
\]

Thus, \( k = \frac{15}{8} \).

Would you like further explanation or details on how inverse functions work?

Here are 5 questions to expand on this topic:
1. What is the process of finding the inverse of a function like \( f(x) = kx^3 \)?
2. How does the concept of one-to-one functions relate to finding the inverse?
3. Can every cubic function have an inverse, and why?
4. What is the significance of the value \( f^{-1}(15) = 2 \) in this context?
5. How would the value of \( k \) change if \( f^{-1}(20) = 2 \) instead?

**Tip:** Remember that a function must be one-to-one (injective) to have an inverse. This ensures that each \( y \)-value corresponds to exactly one \( x \)-value.

Learn how to find the value of k when given the function f(x) = kx^3 and the inverse function f^-1(15) = 2. This problem covers algebraic concepts related to inverse functions and cubic functions, suitable for students in Grades 10-12.

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