Question

The value of \( f(1) \) for \( f \left( \frac{1-x}{1 + x} \right) = x + 2 \) is:

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

To solve functional equations, isolate the function argument and substitute values that simplify the expression.

Answer 1

The Correct Option is B

We are given the functional equation: \[ f\left( \frac{1 - x}{1 + x} \right) = x + 2. \] We need to find \( f(1) \). 
Step 1: Set \( y = \frac{1 - x}{1 + x} \). 
We want to find the value of \( f(1) \). To do this, we first find the value of \( x \) that makes the argument of the function equal to 1. So, we set: \[ \frac{1 - x}{1 + x} = 1. \] Step 2: Solve for \( x \).
To solve for \( x \), we solve the equation: \[ \frac{1 - x}{1 + x} = 1 \quad \Rightarrow \quad 1 - x = 1 + x \quad \Rightarrow \quad -2x = 0 \quad \Rightarrow \quad x = 0. \] Step 3: Substitute \( x = 0 \) into the original equation.
Now that we know \( x = 0 \) makes \( \frac{1 - x}{1 + x} = 1 \), we substitute \( x = 0 \) into the equation \( f\left( \frac{1 - x}{1 + x} \right) = x + 2 \): \[ f(1) = 0 + 2 = 2. \] Thus, the value of \( f(1) \) is \( \boxed{2} \).