To find the inverse function of $f(x) = \frac{8}{1 + x}$, follow these steps:

Step 1: Start with the original function

$y = \frac{8}{1 + x}$

Step 2: Swap $x$ and $y$

To find the inverse, interchange $x$ and $y$: $x = \frac{8}{1 + y}$

Step 3: Solve for $y$

First, multiply both sides by $1 + y$: $x(1 + y) = 8$ Expanding this gives: $x + xy = 8$ Now, isolate $y$: $xy = 8 - x$ $y = \frac{8 - x}{x}$

Step 4: Write the inverse function

The inverse function is: $f^{-1}(x) = \frac{8 - x}{x}$

Final Answer

$f^{-1}(x) = \frac{8 - x}{x}$

Would you like more details or have any questions?

5 Related Questions:

1. How can you verify if two functions are inverses of each other?
2. What is the domain of the inverse function $f^{-1}(x) = \frac{8 - x}{x}$?
3. What happens to the inverse function when $f(x)$ has specific values (e.g., $x = 3$)?
4. How do you find the inverse of more complicated rational functions?
5. How does finding an inverse function differ when dealing with polynomial functions?

Tip

Always check if the function is one-to-one before attempting to find its inverse. This ensures that an inverse function exists.