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

Explanation

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 up the equation
Let \( y = f(x) \). Thus, we have:
\[ y = \frac{e^x - e^{-x}}{e^x + e^{-x}} + 2 \]

### Step 2: Isolate the fraction
Subtract 2 from both sides:
\[ y - 2 = \frac{e^x - e^{-x}}{e^x + e^{-x}} \]

### Step 3: Simplify the fraction
We can rewrite the left side:
\[ y - 2 = \frac{e^x - e^{-x}}{e^x + e^{-x}} = \frac{(e^x - e^{-x})}{(e^x + e^{-x})} \]

### Step 4: Cross-multiply
Cross-multiplying gives:
\[ (y - 2)(e^x + e^{-x}) = e^x - e^{-x} \]

### Step 5: Expand and rearrange
Expanding the left side:
\[ (y - 2)e^x + (y - 2)e^{-x} = e^x - e^{-x} \]
Rearranging gives:
\[ (y - 2)e^x - e^x = - (y - 2)e^{-x} - e^{-x} \]
This simplifies to:
\[ (y - 3)e^x = -(y - 2)e^{-x} \]

### Step 6: Multiply both sides by \( e^x \)
Multiplying both sides by \( e^x \) gives:
\[ (y - 3)e^{2x} = -(y - 2) \]

### Step 7: Solve for \( e^{2x} \)
Rearranging gives:
\[ e^{2x} = -\frac{(y - 2)}{(y - 3)} \]

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

### Step 9: Solve for \( x \)
Dividing by 2 gives:
\[ x = \frac{1}{2} \ln\left(-\frac{(y - 2)}{(y - 3)}\right) \]

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

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