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We are tasked with finding the inverse of the function \( f(x) = (x + 5)^{\frac{1}{7}} \).

### Steps to Find the Inverse:

1. **Start with the given function:**
   \[
   y = (x + 5)^{\frac{1}{7}}
   \]
   We want to solve for \(x\) in terms of \(y\) to find the inverse.

2. **Raise both sides to the power of 7:**
   \[
   y^7 = x + 5
   \]

3. **Solve for \(x\):**
   \[
   x = y^7 - 5
   \]

Thus, the inverse function is:
\[
f^{-1}(x) = x^7 - 5
\]

### Correct Answer:
The correct option is \( f^{-1}(x) = x^7 - 5 \).

Would you like further clarification or a breakdown of any step?

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Here are 5 related questions for further exploration:
1. How do you verify that two functions are inverses of each other?
2. What is the domain and range of the original function \(f(x)\)?
3. How would the inverse change if the exponent were different, say \( \frac{1}{3} \)?
4. What happens to the graph of a function when you find its inverse?
5. Can all functions be inverted? Why or why not?

**Tip:** To check if two functions are inverses, compose them \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \). Both should result in \(x\).

For the function f(x) = (x + 5)^(1/7), find f^(-1)(x).

Learn how to find the inverse of the function f(x) = (x + 5)^(1/7) using algebraic methods. This example covers key concepts such as solving for x and understanding the relationship between exponents and inverse functions. Suitable for students in Grades 10-12.

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