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

### Step 1: Write the function in terms of \( y \)
Let \( y = f(x) \), so:
\[
y = \frac{8}{1 + x}
\]

### Step 2: Solve for \( x \)
Now, solve for \( x \) in terms of \( y \):

1. Multiply both sides by \( 1 + x \) to eliminate the denominator:
   \[
   y(1 + x) = 8
   \]

2. Expand the left side:
   \[
   y + yx = 8
   \]

3. Isolate \( yx \) by subtracting \( y \) from both sides:
   \[
   yx = 8 - y
   \]

4. Solve for \( x \) by dividing both sides by \( y \):
   \[
   x = \frac{8 - y}{y}
   \]

### Step 3: Write the inverse function
Now, replace \( y \) with \( x \) to represent the inverse function:
\[
f^{-1}(x) = \frac{8 - x}{x}
\]

So, the inverse function is:
\[
f^{-1}(x) = \frac{8 - x}{x}
\]

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

Here are some related questions you might find useful:
1. What is the domain of \( f^{-1}(x) \)?
2. How do you verify that \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \)?
3. How does the behavior of \( f(x) \) compare to its inverse?
4. What is the range of the original function \( f(x) \)?
5. Can we determine any asymptotes for both \( f(x) \) and \( f^{-1}(x) \)?

**Tip:** When finding an inverse, always swap \( x \) and \( y \), then solve for the new \( y \). This represents the "undoing" of the original function.

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 step-by-step algebraic process. Understand the key concepts of inverse functions and rational equations, with an example suitable for high school students in Grades 9-12.

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