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

### Step 1: Replace \( f(x) \) with \( y \)
\[
y = \frac{4}{x} - 1
\]

### Step 2: Swap \( x \) and \( y \)
\[
x = \frac{4}{y} - 1
\]

### Step 3: Solve for \( y \)
Add 1 to both sides:
\[
x + 1 = \frac{4}{y}
\]

Now, multiply both sides by \( y \):
\[
y(x + 1) = 4
\]

Finally, solve for \( y \) by dividing both sides by \( x + 1 \):
\[
y = \frac{4}{x + 1}
\]

### Step 4: Replace \( y \) with \( f^{-1}(x) \)
\[
f^{-1}(x) = \frac{4}{x + 1}
\]

So, the inverse function is \( f^{-1}(x) = \frac{4}{x + 1} \).

Would you like more details on this process or have any other questions?

### Related Questions:
1. How do you verify that \( f(x) \) and \( f^{-1}(x) \) are indeed inverses of each other?
2. What is the domain and range of the inverse function \( f^{-1}(x) = \frac{4}{x+1} \)?
3. Can you find the inverse of a function that is not one-to-one? Why or why not?
4. How does the graph of \( f(x) \) compare to the graph of its inverse \( f^{-1}(x) \)?
5. What are the applications of inverse functions in real-world problems?

**Tip:** Always ensure that the original function is one-to-one (passes the horizontal line test) before attempting to find its inverse.

Learn how to find the inverse of the function f(x) = 4/x - 1 with a clear, step-by-step guide. This problem explains the process of swapping x and y, solving for y, and verifying the function's one-to-one property using the horizontal line test. Suitable for students in Grades 10-12.

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