Question

If \( f: \mathbb{R}-\{0\} \to \mathbb{R} \) is defined by \( 3f(x) + 4f\left(\frac{1}{x}\right) = \frac{2-x}{x} \), then \( f(3) = \)

• A. 6
• B. 12
• C. 9
• D. 3

Hint:

In functional equations of the form \( af(x) + bf(1/x) = g(x) \), always replace \( x \) with \( 1/x \) to form a second simultaneous equation. This allows you to solve for \( f(x) \) or \( f(1/x) \) directly using elimination.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:

We are given a functional equation involving \( f(x) \) and \( f\left(\frac{1}{x}\right) \). To find the value of \( f(x) \), we need to eliminate \( f\left(\frac{1}{x}\right) \) by creating a system of linear equations.
Step 2: Key Formula or Approach:

The given equation is: \[ 3f(x) + 4f\left(\frac{1}{x}\right) = \frac{2-x}{x} = \frac{2}{x} - 1 \quad \dots (1) \] Replace \( x \) with \( \frac{1}{x} \) in equation (1) to generate a second equation: \[ 3f\left(\frac{1}{x}\right) + 4f(x) = \frac{2-(1/x)}{1/x} = 2x - 1 \quad \dots (2) \]
Step 3: Detailed Explanation:

Now, we have a system of two equations with variables \( f(x) \) and \( f\left(\frac{1}{x}\right) \): 1. \( 3f(x) + 4f\left(\frac{1}{x}\right) = \frac{2}{x} - 1 \) 2. \( 4f(x) + 3f\left(\frac{1}{x}\right) = 2x - 1 \) Our goal is to find \( f(3) \). We can either solve for \( f(x) \) generally or substitute \( x=3 \) directly into the system. Let's substitute \( x=3 \) directly to simplify calculations. Substitute \( x = 3 \) in equation (1): \[ 3f(3) + 4f\left(\frac{1}{3}\right) = \frac{2-3}{3} = -\frac{1}{3} \quad \dots (A) \] Substitute \( x = 3 \) in equation (2) (which is equivalent to putting \( x=1/3 \) in the original equation): \[ 3f\left(\frac{1}{3}\right) + 4f(3) = 2(3) - 1 = 5 \] Rearranging terms: \[ 4f(3) + 3f\left(\frac{1}{3}\right) = 5 \quad \dots (B) \] Now, solve the system of equations (A) and (B) for \( f(3) \). From (A): Multiply by 3: \[ 9f(3) + 12f\left(\frac{1}{3}\right) = -1 \quad \dots (C) \] From (B): Multiply by 4: \[ 16f(3) + 12f\left(\frac{1}{3}\right) = 20 \quad \dots (D) \] Subtract equation (C) from equation (D): \[ (16f(3) + 12f\left(\frac{1}{3}\right)) - (9f(3) + 12f\left(\frac{1}{3}\right)) = 20 - (-1) \] \[ 7f(3) = 21 \] \[ f(3) = \frac{21}{7} \] \[ f(3) = 3 \]
Step 4: Final Answer:

The value of \( f(3) \) is 3.