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.

Find the inverse function of f(x) = 8 / (1 + x)

Learn how to find the inverse function of f(x) = 8 / (1 + x) through a detailed step-by-step solution. This problem focuses on algebraic manipulation of rational functions, making it suitable for students in Grades 9-11. Explore key concepts in inverse functions and rational equations.

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