If `f(x) = x/(x-1)=1/y` then the value of `f(y)` is

Options

A. x
B. x+1
C. x-1
D. 1-x

Answer

To solve the problem, we start with the given function \( f(x) = \frac{x}{x-1} \) and the equation \( \frac{1}{y} = f(x) \). We need to find the value of \( f(y) \).

### Step-by-Step Solution:

1. **Set up the equation**:
\[ f(x) = \frac{x}{x-1} = \frac{1}{y} \]

2. **Cross-multiply to find y**:
\[ y = \frac{x-1}{x} \]

3. **Substitute y into the function**:
We need to find \( f(y) \). Since \( f(x) = \frac{x}{x-1} \), we can replace \( x \) with \( y \):
\[ f(y) = \frac{y}{y-1} \]

4. **Substitute the expression for y**:
Replace \( y \) with \( \frac{x-1}{x} \):
\[ f(y) = \frac{\frac{x-1}{x}}{\frac{x-1}{x} - 1} \]

5. **Simplify the denominator**:
The denominator becomes:
\[ \frac{x-1}{x} - 1 = \frac{x-1}{x} - \frac{x}{x} = \frac{x-1-x}{x} = \frac{-1}{x} \]

6. **Substitute back into f(y)**:
Now substitute this back into the expression for \( f(y) \):
\[ f(y) = \frac{\frac{x-1}{x}}{\frac{-1}{x}} = \frac{x-1}{-1} = -(x-1) = 1 - x \]

7. **Final result**:
Therefore, the value of \( f(y) \) is:
\[ f(y) = 1 - x \]