Computer the inverse of the function :
`y = (10^(x) - 1)/(10^(x) + 1)`

Answer

To find the inverse of the function \( y = \frac{10^x - 1}{10^x + 1} \), we will follow these steps:

### Step 1: Replace \( y \) with \( x \)
To find the inverse, we start by swapping \( x \) and \( y \):
\[ x = \frac{10^y - 1}{10^y + 1} \]

### Step 2: Multiply both sides by \( 10^y + 1 \)
To eliminate the fraction, multiply both sides by \( 10^y + 1 \):
\[ x(10^y + 1) = 10^y - 1 \]

### Step 3: Distribute \( x \)
Distributing \( x \) gives us:
\[ 10^y \cdot x + x = 10^y - 1 \]

### Step 4: Rearrange the equation
Rearranging the equation to group the \( 10^y \) terms on one side:
\[ 10^y \cdot x - 10^y = -1 - x \]
Factoring out \( 10^y \):
\[ 10^y (x - 1) = -1 - x \]

### Step 5: Solve for \( 10^y \)
Now, isolate \( 10^y \):
\[ 10^y = \frac{-1 - x}{x - 1} \]

### Step 6: Take the logarithm
To solve for \( y \), we take the logarithm base 10 of both sides:
\[ y = \log_{10} \left( \frac{-1 - x}{x - 1} \right) \]

### Step 7: Write the inverse function
Thus, the inverse function is:
\[ f^{-1}(x) = \log_{10} \left( \frac{-1 - x}{x - 1} \right) \]

### Summary
The inverse of the function \( y = \frac{10^x - 1}{10^x + 1} \) is:
\[ f^{-1}(x) = \log_{10} \left( \frac{-1 - x}{x - 1} \right) \]