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To find the inverse function of \( y = e^{x+1} - d \), we need to follow the general steps for determining the inverse of a function. Let's go through the process:

### Given Function:
\[
y = e^{x+1} - d
\]

### Steps to Find the Inverse:

1. **Swap \( x \) and \( y \):**
   This helps us express the inverse function as \( x \) in terms of \( y \):
   \[
   x = e^{y+1} - d
   \]
   
2. **Solve for \( y \):**
   - Add \( d \) to both sides to isolate the exponential term:
     \[
     x + d = e^{y+1}
     \]
   - Take the natural logarithm of both sides to remove the exponential:
     \[
     \ln(x + d) = y + 1
     \]
   - Subtract 1 from both sides to isolate \( y \):
     \[
     y = \ln(x + d) - 1
     \]

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

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### Related Questions:
1. What is the domain of the inverse function?
2. How would you differentiate the inverse function?
3. What is the graph of both the function and its inverse?
4. How does the constant \( d \) affect the inverse function?
5. Can we apply this method to more complex exponential equations?

**Tip:** When working with inverse functions, always check if the original function is one-to-one (invertible).

Learn how to find the inverse of the exponential function y = e^(x+1) - d. This example demonstrates the steps for solving for the inverse using logarithmic properties. Suitable for high school or early college students.

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