If `A=R-{1}` and function `f:AtoA` is defined by `f(x)=(x+1)/(x-1)`, then `f^(-1)(x)` is given by

Options

A. `(1)/(x-1)`
B. `(1)/(1-x)`
C. `(x+1)/(1-x)`
D. `(x+1)/(x-1)`

Answer

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

### Step 1: Set up the equation for the inverse
We start by letting \( y = f(x) \):
\[ y = \frac{x + 1}{x - 1} \]

### Step 2: Solve for \( x \) in terms of \( y \)
To find the inverse, we need to express \( x \) in terms of \( y \). We will cross-multiply to eliminate the fraction:
\[ y(x - 1) = x + 1 \]
This simplifies to:
\[ yx - y = x + 1 \]

### Step 3: Rearrange the equation
Next, we rearrange the equation to isolate terms involving \( x \):
\[ yx - x = y + 1 \]
Factoring out \( x \) from the left side gives:
\[ x(y - 1) = y + 1 \]

### Step 4: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{y + 1}{y - 1} \]

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

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

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