The inverse of the function `f (x) =(e^x -e^(-x))/( e^x +e^(-x) )+2` is given by

Options

A. ` 1/2 log (x-2)/(x-1)`
B. `1/2 log"" (x-1)/(3-x)`
C. `1/2 log ((x)/(2-x))`
D. `-2 log ((x-1)/(1+x))`

Answer

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

### Step 1: Set the function equal to \( y \)
We start by letting \( y = f(x) \):
\[ y = \frac{e^x - e^{-x}}{e^x + e^{-x}} + 2 \]

### Step 2: Isolate the fraction
Next, we isolate the fraction on one side:
\[ y - 2 = \frac{e^x - e^{-x}}{e^x + e^{-x}} \]

### Step 3: Cross-multiply
Now, we cross-multiply to eliminate the fraction:
\[ (y - 2)(e^x + e^{-x}) = e^x - e^{-x} \]

### Step 4: Expand the left side
Expanding the left side gives us:
\[ (y - 2)e^x + (y - 2)e^{-x} = e^x - e^{-x} \]

### Step 5: Rearrange the equation
Rearranging the equation leads to:
\[ (y - 2)e^x - e^x = - (y - 2)e^{-x} - e^{-x} \]
This simplifies to:
\[ ((y - 2) - 1)e^x = -((y - 2) + 1)e^{-x} \]
or
\[ (y - 3)e^x = -(y - 1)e^{-x} \]

### Step 6: Multiply both sides by \( e^x \)
To eliminate \( e^{-x} \), we multiply both sides by \( e^x \):
\[ (y - 3)e^{2x} = -(y - 1) \]

### Step 7: Solve for \( e^{2x} \)
Now, we can isolate \( e^{2x} \):
\[ e^{2x} = \frac{-(y - 1)}{(y - 3)} \]

### Step 8: Take the natural logarithm
Taking the natural logarithm of both sides gives us:
\[ 2x = \ln\left(\frac{-(y - 1)}{(y - 3)}\right) \]

### Step 9: Solve for \( x \)
Finally, we solve for \( x \):
\[ x = \frac{1}{2} \ln\left(\frac{-(y - 1)}{(y - 3)}\right) \]

### Step 10: Write the inverse function
Thus, the inverse function \( f^{-1}(y) \) is:
\[ f^{-1}(y) = \frac{1}{2} \ln\left(\frac{-(y - 1)}{(y - 3)}\right) \]

### Final Answer
The inverse of the function \( f(x) \) is:
\[ f^{-1}(x) = \frac{1}{2} \ln\left(\frac{-(x - 1)}{(x - 3)}\right) \]