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

Options

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

Answer

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

### Step 1: Set \( y = f(x) \)
Let \( y = f(x) = \frac{x+1}{x-1} \).

### Step 2: Solve for \( x \) in terms of \( y \)
We need to express \( x \) in terms of \( y \). Start by cross-multiplying:
\[ y(x - 1) = x + 1 \]
This simplifies to:
\[ yx - y = x + 1 \]

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

### Step 4: Solve for \( x \)
Now, divide both sides by \( (y - 1) \) (assuming \( y \neq 1 \)):
\[ x = \frac{y + 1}{y - 1} \]

### 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 is:
\[ f^{-1}(x) = \frac{x + 1}{x - 1}, \quad x \neq 1 \]

---