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

Options

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

Answer

To solve the problem, we need to evaluate the function \( f(x) = \frac{x-1}{x+1} \) for \( x = -\frac{1}{x} \) and check which of the given options is correct.

### Step-by-step Solution:

1. **Substituting \( x = -\frac{1}{x} \) into \( f(x) \)**:
\[ f\left(-\frac{1}{x}\right) = \frac{-\frac{1}{x} - 1}{-\frac{1}{x} + 1} \]

2. **Simplifying the numerator**:
\[ -\frac{1}{x} - 1 = -\frac{1}{x} - \frac{x}{x} = -\frac{1 + x}{x} \]

3. **Simplifying the denominator**:
\[ -\frac{1}{x} + 1 = -\frac{1}{x} + \frac{x}{x} = \frac{-1 + x}{x} \]

4. **Putting it all together**:
\[ f\left(-\frac{1}{x}\right) = \frac{-\frac{1 + x}{x}}{\frac{-1 + x}{x}} = \frac{-1 - x}{-1 + x} \]

5. **Canceling the negative signs**:
\[ f\left(-\frac{1}{x}\right) = \frac{1 + x}{1 - x} \]

6. **Recognizing the relationship**:
We know that:
\[ f(x) = \frac{x - 1}{x + 1} \]
The reciprocal of \( f(x) \) is:
\[ \frac{1}{f(x)} = \frac{x + 1}{x - 1} \]
Thus, we can see that:
\[ f\left(-\frac{1}{x}\right) = \frac{1 + x}{1 - x} = \frac{1}{f(x)} \]

### Conclusion:
From the calculations, we find that:
\[ f\left(-\frac{1}{x}\right) = \frac{1}{f(x)} \]
This corresponds to option **B**: \( f\left(-\frac{1}{f(x)}\right) = \frac{1}{f(x)} \).