If `f(x)=(x-1)/(x+1)`, then `f^(-1)(x)` is :

Options

A. `(1+x)/(1-x)`
B. `(x+1)/(x-1)`
C. `(1)/(x+1)`
D. none of these

Answer

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

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

### Step 2: Solve for \( x \) in terms of \( y \)
Next, we will rearrange this equation to solve for \( x \). We can cross-multiply to eliminate the fraction:
\[ y(x + 1) = x - 1 \]
Expanding this gives:
\[ yx + y = x - 1 \]

### Step 3: Rearrange the equation
Now, we want to get all terms involving \( x \) on one side and constant terms on the other side:
\[ yx - x = -1 - y \]
Factoring out \( x \) from the left side:
\[ x(y - 1) = -1 - y \]

### Step 4: Solve for \( x \)
Now, we can solve for \( x \) by dividing both sides by \( (y - 1) \):
\[ x = \frac{-1 - y}{y - 1} \]
To simplify this expression, we can multiply the numerator and denominator by -1:
\[ x = \frac{1 + y}{1 - y} \]

### Step 5: Write the inverse function
Since we have expressed \( x \) in terms of \( y \), we can write the inverse function:
\[ f^{-1}(y) = \frac{1 + y}{1 - y} \]
Replacing \( y \) with \( x \) gives us:
\[ f^{-1}(x) = \frac{1 + x}{1 - x} \]

### Conclusion
Thus, the inverse function \( f^{-1}(x) \) is:
\[ f^{-1}(x) = \frac{1 + x}{1 - x} \]

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