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.