To solve \( f^{-1}(16) \) for the function \( f(x) = x^9 \), we need to find the inverse function and then evaluate it at 16.

1. Start by setting \( f(x) = y \), so \( y = x^9 \).
2. To find the inverse, solve for \( x \) in terms of \( y \). Take the 9th root of both sides:
   \[
   x = \sqrt[9]{y}
   \]
   Therefore, the inverse function is:
   \[
   f^{-1}(y) = \sqrt[9]{y}
   \]
3. Now evaluate \( f^{-1}(16) \):
   \[
   f^{-1}(16) = \sqrt[9]{16}
   \]

This means we need to find the 9th root of 16. Using approximate methods:
\[
\sqrt[9]{16} \approx 1.283
\]

So, \( f^{-1}(16) \approx 1.283 \).

Would you like more details on how to approximate the 9th root, or have any other questions?

Here are five related questions to explore further:
1. What is the general method for finding inverse functions?
2. How do we approximate higher-order roots using logarithms?
3. What are other examples of functions with easily found inverses?
4. How would you calculate \( f^{-1}(x) \) for different powers of \( x \)?
5. Can all functions have an inverse? What are the criteria?

**Tip**: Inverse functions "undo" the original function. For a function to have an inverse, it must be one-to-one, meaning it passes the horizontal line test.

This guide walks through solving f(x) = x^9 and evaluating its inverse at f^{-1}(16). Explore the use of inverse functions and the 9th root, with a step-by-step solution and approximation. Suitable for high school students in Grades 9-12.

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