Question

Let \( f \) be a function such that \( f(x) + 3f\left(\frac{24}{x}\right) = 4x \), \( x \neq 0 \). Then \( f(3) + f(8) \) is equal to

• A. 11
• B. 10
• C. 12
• D. 13

Hint:

To solve for the sum of function values at specific points using a functional equation of the form \( af(x) + bf(g(x)) = h(x) \), substitute the specific values and also substitute \( x \) with \( g^{-1}(x) \) (or a related value that connects the arguments of \( f \) in the equation) to create a system of linear equations in terms of the required function values. Solve this system to find the desired sum.

Answer 1

The Correct Option is A

The provided functional equation is:\[f(x) + 3f\left(\frac{24}{x}\right) = 4x\]The objective is to determine the value of \( f(3) + f(8) \).Substituting \( x = 3 \) into the given equation yields:\[f(3) + 3f\left(\frac{24}{3}\right) = 4(3)\]This simplifies to:\[f(3) + 3f(8) = 12 \quad ...(i)\]Next, substituting \( x = 8 \) into the given equation results in:\[f(8) + 3f\left(\frac{24}{8}\right) = 4(8)\]Which simplifies to:\[f(8) + 3f(3) = 32 \quad ...(ii)\]Equations (i) and (ii) form a system of two linear equations with two variables, \( f(3) \) and \( f(8) \).Adding equation (i) and equation (ii) together:\[(f(3) + 3f(8)) + (f(8) + 3f(3)) = 12 + 32\]Combining like terms gives:\[4f(3) + 4f(8) = 44\]Dividing the entire equation by 4:\[f(3) + f(8) = 11\]