Sea f(x)=e2(x−1)−4e(x−1)+b
    con 0<x<1
          y

 g(x)=ln(2+1+x−−−−−√)+1
. Qué valor debe tomar b
 para que f−1(x)=g(x)
 ?

To find the value of \( b \) such that \( f^{-1}(x) = g(x) \), we need to follow these steps:

### 1. **Identify the Functions**:
   - \( f(x) = e^{2(x - 1)} - 4e^{x - 1} + b \)
   - \( g(x) = \ln\left(2 + \sqrt{1 + x}\right) + 1 \)

### 2. **Set the Functions Equal**:
   Since \( f^{-1}(x) = g(x) \), we have:
   \[
   f(g(x)) = x
   \]
   First, express \( f(y) = x \) to find \( y = f^{-1}(x) \):
   \[
   x = e^{2(y - 1)} - 4e^{y - 1} + b
   \]

### 3. **Substitute \( g(x) \) into \( f(y) \)**:
   Substitute \( y = g(x) \) into \( f(y) \) and set the expression equal to \( x \):
   \[
   x = e^{2(g(x) - 1)} - 4e^{g(x) - 1} + b
   \]
   This simplifies to:
   \[
   x = e^{2\left[\ln\left(2 + \sqrt{1 + x}\right)\right]} - 4e^{\ln\left(2 + \sqrt{1 + x}\right)} + b - 4
   \]

### 4. **Simplify the Expression**:
   Simplify the terms involving the exponential functions:
   \[
   x = \left(2 + \sqrt{1 + x}\right)^2 - 4\left(2 + \sqrt{1 + x}\right) + b
   \]
   Expand and combine like terms:
   \[
   x = 4 + 4\sqrt{1 + x} + 1 + x - 8 - 4\sqrt{1 + x} + b
   \]
   This simplifies to:
   \[
   x = x + b - 3
   \]
   Hence, \( b - 3 = 0 \), so \( b = 3 \).

### **Conclusion**:
The value of \( b \) that makes \( f^{-1}(x) = g(x) \) is \( \boxed{3} \).

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Would you like to go through the detailed steps or have any questions?

Here are 5 related questions:

1. How do you find the inverse of a function algebraically?
2. What are the properties of exponential and logarithmic functions?
3. Can you solve for \( b \) if the domain of \( f(x) \) changes?
4. What is the significance of the condition \( 0 < x < 1 \) for the function \( f(x) \)?
5. How does the derivative of a function relate to finding its inverse?

**Tip:** When finding the inverse of a function, ensure that the function is one-to-one by checking its monotonicity (increasing or decreasing behavior).

Learn how to find the value of b to satisfy the condition f^-1(x) = g(x) with detailed step-by-step calculations. This problem involves exponential and logarithmic functions, exploring their properties and inverse relationships. Suitable for high school students studying advanced algebra and functions.

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