Question

If \( f(x) = \frac{x-1}{x+1} \), then which of the following will be true?

• A. \( f\left( \frac{1}{x} \right) = f(x) \)
• B. \( f\left( \frac{-1}{x} \right) = -f(x) \)
• C. \( f\left( \frac{1}{x} \right) = -f(x) \)
• D. \( f\left( \frac{-1}{x} \right) = \frac{1}{f(x)} \)

Hint:

For such functional questions, substitute the expression into the function and simplify to check if the equation holds true.

Answer 1

The Correct Option is B

We are given the function \( f(x) = \frac{x-1}{x+1} \). Let's evaluate \( f\left( \frac{-1}{x} \right) \): \[ f\left( \frac{-1}{x} \right) = \frac{\frac{-1}{x} - 1}{\frac{-1}{x} + 1} = \frac{\frac{-1 - x}{x}}{\frac{-1 + x}{x}} = \frac{-1 - x}{-1 + x} = -\frac{x + 1}{x - 1} = -f(x) \] Therefore, the correct option is (2).