Question

Let \(f :R→R\) be a continuous function such that \(f(3x) – f(x) = x\). If \(f(8) = 7\), then \(f(14\)) is equal to

• A. 4
• B. 10
• C. 11
• D. 18

Answer 1

The Correct Option is B

1. We are given the functional equation \( f(3x) - f(x) = x \). To find a particular solution, let's analyze and find \( f(x) \).
2. Let's start by trying specific values to understand the pattern:
   • For \( x = 1 \): \( f(3) - f(1) = 1 \)
   • For \( x = 3 \): \( f(9) - f(3) = 3 \)
   • For \( x = 9 \): \( f(27) - f(9) = 9 \)
   From these examples, we deduce: \[ f(3) = f(1) + 1 \] \[ f(9) = f(3) + 3 = f(1) + 1 + 3 = f(1) + 4 \] \[ f(27) = f(9) + 9 = f(1) + 4 + 9 = f(1) + 13 \]
3. This suggests a general approach. Assume \( f(x) = x + c \) where \( c \) is a constant. Substitute back: \[ f(3x) = 3x + c \] \[ f(3x) - f(x) = (3x + c) - (x + c) = 2x \] However, we have \( f(3x) - f(x) = x \), which means the assumption \( f(x) = x + c \) does not satisfy the given functional equation with this analysis.
4. To find an appropriate solution, observe the simplified logic if \( c \) can branch another pattern for constant: \[ f(3x) = f(x) + x \] Recursively applying yields: \[ f(x) = \frac{x}{2} + c \]
5. We know \( f(8) = 7 \) \[ f(8) = \frac{8}{2} + c = 4 + c = 7 \] Solving above, \[ c = 3 \]
6. Therefore, \( f(x) = \frac{x}{2} + 3 \). To find \( f(14) \): \[ f(14) = \frac{14}{2} + 3 = 7 + 3 = 10 \]
7. Thus, the answer is 10.