Determine `f^(-1)(x)`, if given function is invertible.
`f:(-oo,1)to(-oo,-2)` defined by `f(x)=-(x+1)^(2)-2`

Explanation

To find the inverse of the function \( f(x) = -(x+1)^2 - 2 \), we will follow these steps:

### Step 1: Set the function equal to \( y \)
Let \( y = f(x) \). Thus, we have:
\[ y = -(x + 1)^2 - 2 \]

### Step 2: Solve for \( x \)
We need to isolate \( x \) in terms of \( y \). First, we can rearrange the equation:
\[ y + 2 = -(x + 1)^2 \]
Next, multiply both sides by -1:
\[ -(y + 2) = (x + 1)^2 \]
This simplifies to:
\[ (x + 1)^2 = -y - 2 \]

### Step 3: Take the square root
Now, we take the square root of both sides. Remember to consider both the positive and negative roots, but since we are given the domain of \( f \), we will only take the principal root:
\[ x + 1 = \sqrt{-y - 2} \]
Thus,
\[ x = \sqrt{-y - 2} - 1 \]

### Step 4: Express \( f^{-1}(x) \)
Now we can express the inverse function \( f^{-1}(x) \) by substituting \( y \) with \( x \):
\[ f^{-1}(x) = \sqrt{-x - 2} - 1 \]

### Final Answer
The inverse function is:
\[ f^{-1}(x) = \sqrt{-x - 2} - 1 \]

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