To find the inverse of the function \( f(x) = \frac{-2}{x+1} - 2 \), follow these steps:

### Step 1: Replace \( f(x) \) with \( y \)
\[
y = \frac{-2}{x+1} - 2
\]

### Step 2: Swap \( y \) and \( x \)
To find the inverse, we switch the roles of \( x \) and \( y \):
\[
x = \frac{-2}{y+1} - 2
\]

### Step 3: Solve for \( y \)
Now, we need to solve for \( y \) in terms of \( x \).

1. First, add 2 to both sides:
   \[
   x + 2 = \frac{-2}{y+1}
   \]

2. Multiply both sides by \( (y+1) \) to eliminate the denominator:
   \[
   (x + 2)(y + 1) = -2
   \]

3. Expand the left side:
   \[
   (x + 2)(y + 1) = x(y + 1) + 2(y + 1) = -2
   \]
   Simplifying:
   \[
   (x + 2)y + (x + 2) = -2
   \]

4. Subtract \( (x + 2) \) from both sides:
   \[
   (x + 2)y = -2 - (x + 2)
   \]
   Simplifying:
   \[
   (x + 2)y = -x - 4
   \]

5. Finally, divide by \( (x + 2) \):
   \[
   y = \frac{-x - 4}{x + 2}
   \]

Thus, the inverse function is:
\[
f^{-1}(x) = \frac{-x - 4}{x + 2}
\]

Would you like more details or have any further questions?

Here are 5 related questions to consider:
1. How can you verify if a function and its inverse are correct?
2. What are the domain and range of the inverse function?
3. How do we find the inverse of a more complex function?
4. What is the graphical relationship between a function and its inverse?
5. How does the method change if the function involves powers or roots?

**Tip:** To verify if two functions are inverses, compose them: \( f(f^{-1}(x)) \) and \( f^{-1}(f(x)) \) should both return \( x \).

Learn how to find the inverse of the rational function f(x) = (-2/(x+1)) - 2 with a detailed, step-by-step explanation. This solution involves algebraic manipulation and the inverse function formula, making it suitable for students in Grades 10-12.

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