To find the inverse of the function \( f(x) = \frac{1}{x} + 13 \), we follow these steps:

1. Replace \( f(x) \) with \( y \):
   \[
   y = \frac{1}{x} + 13
   \]

2. Swap \( y \) and \( x \) to set up the equation for the inverse function:
   \[
   x = \frac{1}{y} + 13
   \]

3. Isolate \( y \) by subtracting 13 from both sides:
   \[
   x - 13 = \frac{1}{y}
   \]

4. Take the reciprocal of both sides to solve for \( y \):
   \[
   y = \frac{1}{x - 13}
   \]

Thus, the inverse function is:
\[
f^{-1}(x) = \frac{1}{x - 13}
\]

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you verify if two functions are inverses of each other?
2. What is the domain of \( f^{-1}(x) \)?
3. How does the graph of \( f(x) \) compare to the graph of \( f^{-1}(x) \)?
4. How do inverse functions affect compositions, like \( f(f^{-1}(x)) \)?
5. How does the transformation \( +13 \) in the original function affect the inverse?

**Tip**: When finding the inverse, always make sure to check the domain of both the original and inverse functions to avoid undefined values.

This problem guides you through finding the inverse of the function f(x) = 1/x + 13. It involves key algebraic concepts like inverse functions and the manipulation of expressions. Ideal for students in Grades 9-12 who are studying functions and algebra.

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