To find the inverse function \( f^{-1}(x) \) of the given function \( f(x) = \frac{1}{x} + 7 \), we need to follow these steps:

### Step 1: Replace \( f(x) \) with \( y \)
Let \( y = f(x) = \frac{1}{x} + 7 \).

\[
y = \frac{1}{x} + 7
\]

### Step 2: Solve for \( x \)
We want to express \( x \) in terms of \( y \). First, subtract 7 from both sides:

\[
y - 7 = \frac{1}{x}
\]

Next, take the reciprocal of both sides:

\[
x = \frac{1}{y - 7}
\]

### Step 3: Replace \( y \) with \( x \)
Now, replace \( y \) with \( x \) to get the inverse function:

\[
f^{-1}(x) = \frac{1}{x - 7}
\]

Thus, the inverse function is:

\[
f^{-1}(x) = \frac{1}{x - 7}
\]

Would you like more details on the process, or do you have any questions?

Here are 5 related questions to explore:
1. What are the domain and range of the original function \( f(x) = \frac{1}{x} + 7 \)?
2. How can we verify that \( f(f^{-1}(x)) = x \) for this function?
3. What are the domain and range of the inverse function \( f^{-1}(x) \)?
4. How does shifting a function vertically (like adding 7) affect its inverse?
5. What happens to the graph of \( f(x) = \frac{1}{x} + 7 \) compared to \( f(x) = \frac{1}{x} \)?

**Tip**: When finding the inverse of a function, always check that the resulting inverse undoes the operations of the original function.

Learn how to find the inverse function of f(x) = 1/x + 7. This step-by-step guide walks through solving rational functions and applying the inverse function formula. Ideal for students in Grades 10-12 exploring advanced algebra concepts.

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