Key Concepts and Formulas

• Functional Equations: Understanding how to manipulate and solve equations involving functions.
• Geometric Series: The sum of an infinite geometric series $a + ar + ar^2 + \dots$ is given by $\frac{a}{1-r}$ for $|r| < 1$.
• Continuity of Functions: If a function $f$ is continuous at a point $c$, then $\lim_{x \to c} f(x) = f(c)$.

Step-by-Step Solution

Step 1: Analyze the given functional equation. We are given the functional equation $f(3x) - f(x) = x$ for all $x \in \mathbb{R}$. This equation relates the function's value at a point $x$ to its value at $3x$.

Step 2: Generate a sequence of related equations by repeatedly substituting $x$ with $x/3$. The goal is to create a telescoping sum. Given $f(3x) - f(x) = x$. Let $x$ be replaced by $x/3$: $f(3(x/3)) - f(x/3) = x/3$ $f(x) - f(x/3) = x/3$ (Equation 2)

Let $x/3$ be replaced by $x/9$ in Equation 2: $f(x/3) - f(x/9) = (x/3)/3 = x/9$ (Equation 3)

Continuing this pattern, for any positive integer $n$, we can write: $f\left(\frac{x}{3^{k-1}}\right) - f\left(\frac{x}{3^k}\right) = \frac{x}{3^k}$ for $k = 1, 2, \dots, n$.

Let's write out the first few terms and the general term for $n$ terms: For $k=1$: $f(x) - f(x/3) = x/3$ For $k=2$: $f(x/3) - f(x/9) = x/9$ For $k=3$: $f(x/9) - f(x/27) = x/27$ ... For $k=n$: $f\left(\frac{x}{3^{n-1}}\right) - f\left(\frac{x}{3^n}\right) = \frac{x}{3^n}$

Step 3: Sum these equations to form a telescoping series. Adding the equations from $k=1$ to $k=n$: $(f(x) - f(x/3)) + (f(x/3) - f(x/9)) + \dots + \left(f\left(\frac{x}{3^{n-1}}\right) - f\left(\frac{x}{3^n}\right)\right) = \frac{x}{3} + \frac{x}{9} + \dots + \frac{x}{3^n}$

The left side is a telescoping sum, where intermediate terms cancel out: $f(x) - f\left(\frac{x}{3^n}\right) = x\left(\frac{1}{3} + \frac{1}{9} + \dots + \frac{1}{3^n}\right)$

The right side is a finite geometric series with first term $a = 1/3$, common ratio $r = 1/3$, and $n$ terms. The sum is $S_n = a\frac{1-r^n}{1-r} = \frac{1}{3}\frac{1-(1/3)^n}{1-1/3} = \frac{1}{3}\frac{1-(1/3)^n}{2/3} = \frac{1}{2}\left(1 - \frac{1}{3^n}\right)$. So, $f(x) - f\left(\frac{x}{3^n}\right) = x \cdot \frac{1}{2}\left(1 - \frac{1}{3^n}\right)$.

Step 4: Take the limit as $n \to \infty$ using the continuity of $f$. As $n \to \infty$, the term $\frac{x}{3^n} \to 0$ (assuming $x$ is finite). Since $f$ is continuous on $\mathbb{R}$, we have $\lim_{n \to \infty} f\left(\frac{x}{3^n}\right) = f\left(\lim_{n \to \infty} \frac{x}{3^n}\right) = f(0)$. Also, as $n \to \infty$, $\frac{1}{3^n} \to 0$.

Taking the limit of the equation from Step 3: $\lim_{n \to \infty} \left( f(x) - f\left(\frac{x}{3^n}\right) \right) = \lim_{n \to \infty} \left( \frac{x}{2}\left(1 - \frac{1}{3^n}\right) \right)$ $f(x) - f(0) = \frac{x}{2}(1 - 0)$ $f(x) - f(0) = \frac{x}{2}$

This gives us a general form for the function: $f(x) = \frac{x}{2} + C$, where $C = f(0)$ is a constant.

Step 5: Use the given condition $f(8) = 7$ to find the constant $C$. Substitute $x=8$ into the general form: $f(8) = \frac{8}{2} + C$ $7 = 4 + C$ $C = 7 - 4 = 3$. So, $f(0) = 3$.

The function is $f(x) = \frac{x}{2} + 3$.

Step 6: Calculate $f(14)$ using the derived function. Substitute $x=14$ into the function $f(x) = \frac{x}{2} + 3$: $f(14) = \frac{14}{2} + 3$ $f(14) = 7 + 3$ $f(14) = 10$.

Alternative approach using the original functional equation directly to find $f(0)$: From Step 4, we have $f(x) - f(0) = x/2$. This implies $f(x) = x/2 + f(0)$. We are given $f(8) = 7$. Substituting $x=8$: $f(8) = 8/2 + f(0) = 4 + f(0)$. So, $7 = 4 + f(0)$, which gives $f(0) = 3$. Now, we need to find $f(14)$. Using the relation $f(x) - f(0) = x/2$, substitute $x=14$: $f(14) - f(0) = 14/2$ $f(14) - 3 = 7$ $f(14) = 7 + 3 = 10$.

Common Mistakes & Tips

• Incorrect Geometric Series Sum: Ensure the formula for the sum of an infinite geometric series is applied correctly, especially the condition for convergence ($|r| < 1$).
• Algebraic Errors in Telescoping Sum: Carefully cancel out terms in the telescoping sum to avoid calculation mistakes.
• Misinterpreting Continuity: Remember that continuity allows us to move the limit inside the function: $\lim_{n \to \infty} f(a_n) = f(\lim_{n \to \infty} a_n)$ if $f$ is continuous.

Summary

We started with the given functional equation and generated a sequence of equations by repeatedly substituting $x$ with $x/3$. By summing these equations, we obtained a telescoping series that simplified to $f(x) - f(x/3^n) = x(1/3 + 1/9 + \dots + 1/3^n)$. Taking the limit as $n \to \infty$ and using the continuity of $f$, we derived the form $f(x) - f(0) = x/2$, which means $f(x) = x/2 + C$. Using the given condition $f(8) = 7$, we found the constant $C=3$. Finally, we used the derived function $f(x) = x/2 + 3$ to calculate $f(14) = 10$.

The final answer is $\boxed{10}$.