Question

If \[ f(x) + 3f(g(x)) = x - 2, \] where \[ g(x) = \frac{3x + 1}{x - 3}, \] then the value of the ratio \[ \frac{f(5)}{f(8)} \text{ is } _______ \text{ (answer in integer).}

Hint:

To solve functional equations, substitute specific values for \( x \) to simplify and solve for the unknowns.

Answer 1

Correct Answer: 5

First, let's solve for \( f(x) \) using the given equation. We know: \[ f(x) + 3f(g(x)) = x - 2. \] Substitute \( g(x) \) into the equation: \[ f(x) + 3f\left( \frac{3x + 1}{x - 3} \right) = x - 2. \] Now, substitute \( x = 5 \) and \( x = 8 \) into the equation to find \( f(5) \) and \( f(8) \). After calculations, we find the ratio: \[ \frac{f(5)}{f(8)} = 5. \] Thus, the value of the ratio is \( 5 \).