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 given functional equation is: \[ f(x) + 3f\left(\frac{24}{x}\right) = 4x \] We need to find the value of \( f(3) + f(8) \). Substitute \( x = 3 \) in the given equation: \[ f(3) + 3f\left(\frac{24}{3}\right) = 4(3) \] \[ f(3) + 3f(8) = 12 \quad ...(i) \] Substitute \( x = 8 \) in the given equation: \[ f(8) + 3f\left(\frac{24}{8}\right) = 4(8) \] \[ f(8) + 3f(3) = 32 \quad ...(ii) \] We have a system of two linear equations with two unknowns, \( f(3) \) and \( f(8) \). Adding equation (i) and equation (ii): \[ (f(3) + 3f(8)) + (f(8) + 3f(3)) = 12 + 32 \] \[ 4f(3) + 4f(8) = 44 \] Divide by 4: \[ f(3) + f(8) = 11 \]

Answer 2

Let's solve the given functional equation: \( f(x) + 3f\left(\frac{24}{x}\right) = 4x \), where \( x \neq 0 \).

1. Substitute \( x = a \) in the equation. We get: \(f(a) + 3f\left(\frac{24}{a}\right) = 4a\) (Equation 1)
2. Substitute \( x = \frac{24}{a} \) in the equation. We get: \(f\left(\frac{24}{a}\right) + 3f(a) = \frac{96}{a}\) (Equation 2)
3. Now, we have two equations:
   1. \(f(a) + 3f\left(\frac{24}{a}\right) = 4a\)
   2. \(f\left(\frac{24}{a}\right) + 3f(a) = \frac{96}{a}\)
4. Let's multiply Equation 1 by 3 and subtract Equation 2 from it:
   1. After multiplying Equation 1 by 3: \(3f(a) + 9f\left(\frac{24}{a}\right) = 12a\)
   2. Subtract Equation 2: \((3f(a) + 9f\left(\frac{24}{a}\right)) - (f\left(\frac{24}{a}\right) + 3f(a)) = 12a - \frac{96}{a}\)
   3. This simplifies to: \(8f\left(\frac{24}{a}\right) = 12a - \frac{96}{a}\)
   4. Solving, we get: \(f\left(\frac{24}{a}\right) = \frac{12a - \frac{96}{a}}{8} = \frac{6a^2 - 48}{4a}\)
   5. This simplifies to: \(f\left(\frac{24}{a}\right) = \frac{3a - \frac{48}{a}}{4}\)
5. The equation for \( f(x) \) becomes: \(f(x) = \frac{3x - \frac{48}{x}}{4}\)
6. Now we need to find \( f(3) + f(8) \):
   1. Calculate \( f(3) \): \(f(3) = \frac{3 \times 3 - \frac{48}{3}}{4} = \frac{9 - 16}{4} = -\frac{7}{4}\)
   2. Calculate \( f(8) \): \(f(8) = \frac{3 \times 8 - \frac{48}{8}}{4} = \frac{24 - 6}{4} = \frac{18}{4} = \frac{9}{2}\)
   3. Finally, compute \( f(3) + f(8) \): \(f(3) + f(8) = -\frac{7}{4} + \frac{9}{2} = \frac{-7 + 18}{4} = \frac{11}{4} = 11\)
7. Thus, the answer is 11.