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To find the inverse of the function \( f(x) = (x + 3)^5 + 9 \), we follow these steps:

1. **Write the function in terms of \( y \):**

   \[
   y = (x + 3)^5 + 9
   \]

2. **Isolate the expression involving \( x \):**

   First, subtract 9 from both sides:

   \[
   y - 9 = (x + 3)^5
   \]

3. **Take the 5th root of both sides to solve for \( x + 3 \):**

   \[
   \sqrt[5]{y - 9} = x + 3
   \]

4. **Solve for \( x \):**

   Subtract 3 from both sides:

   \[
   x = \sqrt[5]{y - 9} - 3
   \]

5. **Rewrite the inverse function:**

   Thus, the inverse function \( f^{-1}(x) \) is:

   \[
   f^{-1}(x) = \sqrt[5]{x - 9} - 3
   \]

Let me know if you would like more details or if you have any questions!

Here are five related questions:

1. How can we verify that \( f^{-1}(x) \) is indeed the inverse of \( f(x) \)?
2. What is the domain and range of the inverse function \( f^{-1}(x) \)?
3. How does the graph of \( f(x) \) compare to the graph of \( f^{-1}(x) \)?
4. Can we determine the derivative of \( f^{-1}(x) \)?
5. What would the inverse look like if the exponent were different (e.g., 3 or 4)?

**Tip:** To check if two functions are inverses of each other, compose them \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \); both should return \( x \).

Learn how to find the inverse of the function f(x) = (x + 3)^5 + 9 with a step-by-step solution. This problem involves key concepts like inverse functions and solving for variables. Suitable for students in Grades 11-12.

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