f (x) is any function and `f^(-1)(x)` is known as inverse of f(x), then `f^(-1)(x)` of `f(x)= x//(x-1), x ne 1` is

Options

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

Answer

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

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

### Step 2: Rearrange the equation to solve for \( x \)
We need to express \( x \) in terms of \( y \). Start by multiplying both sides by \( (x - 1) \) to eliminate the fraction:
\[ y(x - 1) = x \]
This simplifies to:
\[ yx - y = x \]

### Step 3: Collect all terms involving \( x \) on one side
Rearranging gives:
\[ yx - x = y \]
Factoring out \( x \) from the left side:
\[ x(y - 1) = y \]

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

### Step 5: Replace \( y \) with \( x \) to find \( f^{-1}(x) \)
Now, we replace \( y \) with \( x \) to express the inverse function:
\[ f^{-1}(x) = \frac{x}{x - 1} \]

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

---