Key Concepts and Formulas

• Functional Equations: Equations involving an unknown function and its values at various arguments.
• Substitution Method for Functional Equations: When a functional equation has a structure like $f(x) + a f(g(x)) = h(x)$ and $g(g(x)) = x$, substituting $g(x)$ for $x$ generates a second equation. This pair of equations can be treated as a system of linear equations in terms of $f(x)$ and $f(g(x))$.
• System of Linear Equations: A set of two or more linear equations with the same variables. Can be solved using methods like elimination or substitution.

Step-by-Step Solution

Step 1: Understand the Given Functional Equation We are given the functional equation $f(x)+3 f\left(\frac{24}{x}\right)=4 x$, where $x \neq 0$. This equation relates the function's value at $x$ to its value at $\frac{24}{x}$. Our objective is to find the value of $f(3)+f(8)$.

Step 2: Apply Strategic Substitution The structure of the equation suggests substituting $\frac{24}{x}$ for $x$. Let $g(x) = \frac{24}{x}$. Notice that $g(g(x)) = g\left(\frac{24}{x}\right) = \frac{24}{\frac{24}{x}} = x$. This property is key. Substitute $x$ with $\frac{24}{x}$ in the given equation: $f\left(\frac{24}{x}\right) + 3 f\left(\frac{24}{\frac{24}{x}}\right) = 4\left(\frac{24}{x}\right)$ Simplifying the argument of the second $f$ term: $f\left(\frac{24}{x}\right) + 3 f(x) = \frac{96}{x}$ Let's call this the second equation: $3 f(x) + f\left(\frac{24}{x}\right) = \frac{96}{x} \quad \ldots(2)$

Step 3: Form a System of Linear Equations We now have a system of two linear equations involving $f(x)$ and $f\left(\frac{24}{x}\right)$: Equation (1): $f(x) + 3 f\left(\frac{24}{x}\right) = 4x$ Equation (2): $3 f(x) + f\left(\frac{24}{x}\right) = \frac{96}{x}$

Step 4: Solve the System for $f(x)$ To find an explicit expression for $f(x)$, we can eliminate $f\left(\frac{24}{x}\right)$. Multiply Equation (2) by 3: $3 \times \left(3 f(x) + f\left(\frac{24}{x}\right)\right) = 3 \times \left(\frac{96}{x}\right)$ $9 f(x) + 3 f\left(\frac{24}{x}\right) = \frac{288}{x} \quad \ldots(3)$ Now, subtract Equation (1) from Equation (3): $(9 f(x) + 3 f\left(\frac{24}{x}\right)) - (f(x) + 3 f\left(\frac{24}{x}\right)) = \frac{288}{x} - 4x$ The terms involving $f\left(\frac{24}{x}\right)$ cancel out: $8 f(x) = \frac{288}{x} - 4x$ Divide by 8 to isolate $f(x)$: $f(x) = \frac{1}{8} \left(\frac{288}{x} - 4x\right)$ $f(x) = \frac{36}{x} - \frac{x}{2}$

Step 5: Calculate $f(3)$ and $f(8)$ Using the derived formula for $f(x)$: For $f(3)$: $f(3) = \frac{36}{3} - \frac{3}{2} = 12 - \frac{3}{2}$ For $f(8)$: $f(8) = \frac{36}{8} - \frac{8}{2} = \frac{9}{2} - 4$

Step 6: Compute $f(3) + f(8)$ Now, add the values of $f(3)$ and $f(8)$: $f(3) + f(8) = \left(12 - \frac{3}{2}\right) + \left(\frac{9}{2} - 4\right)$ Group the integer and fractional parts: $f(3) + f(8) = (12 - 4) + \left(\frac{9}{2} - \frac{3}{2}\right)$ $f(3) + f(8) = 8 + \frac{6}{2}$ $f(3) + f(8) = 8 + 3$ $f(3) + f(8) = 11$

Common Mistakes & Tips

• Incorrect Substitution: Ensure that every instance of $x$ is replaced by $\frac{24}{x}$ in the original equation.
• Algebraic Errors: Be meticulous with fraction arithmetic and signs when solving the system of equations. A small mistake can lead to an incorrect final answer.
• Verification: After finding $f(x)$, it's a good practice to substitute it back into the original functional equation to confirm its correctness.

Summary The problem involves solving a functional equation by recognizing that the substitution of $\frac{24}{x}$ for $x$ leads to a system of linear equations. By forming this system and solving for $f(x)$, we obtained the explicit form $f(x) = \frac{36}{x} - \frac{x}{2}$. Evaluating this function at $x=3$ and $x=8$ and summing the results gives $f(3) + f(8) = 11$.

Final Answer The final answer is $\boxed{11}$.