To find the inverse of the function \( f(x) = \frac{1}{x} + 4 \), we need to follow these steps:

1. **Replace \( f(x) \) with \( y \):**
   \[
   y = \frac{1}{x} + 4
   \]

2. **Swap \( x \) and \( y \) to solve for the inverse:**
   \[
   x = \frac{1}{y} + 4
   \]

3. **Solve for \( y \):**
   \[
   x - 4 = \frac{1}{y}
   \]
   \[
   y = \frac{1}{x - 4}
   \]

So, the inverse function is:
\[
f^{-1}(x) = \frac{1}{x - 4}
\]

Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:
1. How do you verify that \( f(f^{-1}(x)) = x \)?
2. What is the domain and range of \( f^{-1}(x) = \frac{1}{x - 4} \)?
3. How do you find the inverse of a more complicated rational function?
4. Can all functions have an inverse? If not, why?
5. How do transformations (like shifting or scaling) affect the inverse of a function?

**Tip:** The domain of an inverse function is the range of the original function, and vice versa. Keep this in mind when solving inverse problems!

Let f(x) = 1/x + 4, what does f^-1(x) = ?

Explore the step-by-step process to find the inverse of the function f(x) = 1/x + 4. This solution includes key algebraic concepts, rational functions, and the inverse function theorem. Ideal for students in Grades 10-12.

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