If f (x) is a function satisfying `f (x). f (1//x) = f (x) + f (1//x)` and f (4) = 65, what will be the value of f (6)?

Options

A. 37
B. 217
C. 64
D. None of these

Answer

To solve the problem, we start with the functional equation given:

1. **Functional Equation**:
\[ f(x) \cdot f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right) \]

2. **Rearranging the Equation**:
We can rearrange the equation as follows:
\[ f(x) \cdot f\left(\frac{1}{x}\right) - f(x) - f\left(\frac{1}{x}\right) = 0 \]
This can be factored as:
\[ (f(x) - 1)(f\left(\frac{1}{x}\right) - 1) = 1 \]

3. **Substituting Values**:
We are given that \( f(4) = 65 \). We can use this to find \( f\left(\frac{1}{4}\right) \):
\[ (f(4) - 1)(f\left(\frac{1}{4}\right) - 1) = 1 \]
Substituting \( f(4) = 65 \):
\[ (65 - 1)(f\left(\frac{1}{4}\right) - 1) = 1 \]
Simplifying:
\[ 64(f\left(\frac{1}{4}\right) - 1) = 1 \]
Therefore:
\[ f\left(\frac{1}{4}\right) - 1 = \frac{1}{64} \]
Thus:
\[ f\left(\frac{1}{4}\right) = 1 + \frac{1}{64} = \frac{65}{64} \]

4. **Finding a General Form for \( f(x) \)**:
Based on the functional equation, we can assume a form for \( f(x) \):
\[ f(x) = x^n + 1 \]
We can verify this by substituting back into the original equation.

5. **Using the Given Value**:
Since \( f(4) = 65 \):
\[ 4^n + 1 = 65 \]
This simplifies to:
\[ 4^n = 64 \]
Thus:
\[ 4^n = 4^3 \implies n = 3 \]

6. **Final Function**:
Therefore, we have:
\[ f(x) = x^3 + 1 \]

7. **Finding \( f(6) \)**:
Now we need to find \( f(6) \):
\[ f(6) = 6^3 + 1 = 216 + 1 = 217 \]

Thus, the value of \( f(6) \) is \( \boxed{217} \).