Question

If $y = f(x)$ and $f(x) = \dfrac{1 - x}{1 + x}$, which of the following is true?

• A. $f(2x) = f(x) - 1$
• B. $x = f(2y) - 1$
• C. $f(1/x) = f(x)$
• D. $x = f(y)$

Hint:

When given a rational function and asked to compare values involving inverses, try expressing one variable in terms of the other by solving algebraically.

Answer 1

The Correct Option is D

We are given: \[ y = f(x) = \frac{1 - x}{1 + x} \] Step 1: Express $x$ in terms of $y$ (invert the function) We start with: \[ y = \frac{1 - x}{1 + x} \] Multiply both sides by $(1 + x)$: \[ y(1 + x) = 1 - x \Rightarrow y + yx = 1 - x \] Bring all terms to one side: \[ y + yx + x - 1 = 0 \Rightarrow x(y + 1) = 1 - y \Rightarrow x = \frac{1 - y}{y + 1} \] So we have: \[ x = f(y) \] \[ \boxed{x = f(y)} \]