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The given equation is:

\[
y = e^{x+1} - 2
\]

To find the inverse function of this equation, we need to follow these steps:

### Step 1: Express \( x \) in terms of \( y \).
We start by solving for \( x \).

1. Add 2 to both sides:
\[
y + 2 = e^{x+1}
\]

2. Take the natural logarithm (ln) of both sides to get rid of the exponential:
\[
\ln(y + 2) = x + 1
\]

3. Subtract 1 from both sides to isolate \( x \):
\[
x = \ln(y + 2) - 1
\]

### Step 2: Express the inverse function.
Now, we can rewrite the equation as:
\[
x = \ln(y + 2) - 1
\]

Swapping \( x \) and \( y \) to express the inverse function:
\[
y = \ln(x + 2) - 1
\]

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

### Summary:
The inverse of the given function \( y = e^{x+1} - 2 \) is \( f^{-1}(x) = \ln(x + 2) - 1 \).

Would you like more details or have any questions?

### Related Questions:
1. How can you verify if a function and its inverse are correct?
2. What is the domain and range of the original function \( y = e^{x+1} - 2 \)?
3. What is the domain and range of the inverse function \( f^{-1}(x) \)?
4. How do natural logarithms help in solving equations involving exponentials?
5. What are the key properties of exponential and logarithmic functions?

### Tip:
To confirm that two functions are inverses, you can compose them: if \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \), they are indeed inverses.

Learn how to find the inverse of the exponential function y = e^{x+1} - 2 with a detailed step-by-step solution. This problem involves key mathematical concepts such as inverse functions, exponential equations, and logarithms. Suitable for high school students in Grades 10-12.

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