If f(x) is a polynomial function satisfying the condition `f(x) .f((1)/(x)) = f(x) + f((1)/(x))` and f(2) = 9 then

Options

A. 2f(4) = 3f(6)
B. 14f(1) = f(3)
C. 9f(3) = 2f(5)
D. f(10) = f(11)

Answer

To solve the problem, we need to find the polynomial function \( f(x) \) that satisfies the equation:

\[ f(x) \cdot f\left(\frac{1}{x}\right) = f(x) + f\left(\frac{1}{x}\right) \]

and is given that \( f(2) = 9 \).

### Step 1: Analyze the functional equation

The equation can be rearranged to:

\[ f(x) \cdot f\left(\frac{1}{x}\right) - f(x) - f\left(\frac{1}{x}\right) = 0 \]

This suggests that \( f(x) \) and \( f\left(\frac{1}{x}\right) \) are related in a specific way.

### Step 2: Assume a form for \( f(x) \)

We can assume a polynomial form for \( f(x) \). A common approach is to try:

\[ f(x) = a + b x^n \]

for some constants \( a \), \( b \), and \( n \).

### Step 3: Substitute and simplify

Substituting \( f(x) = a + b x^n \) into the functional equation:

\[ f\left(\frac{1}{x}\right) = a + b \left(\frac{1}{x}\right)^n = a + \frac{b}{x^n} \]

Now substituting into the original equation:

\[ (a + b x^n) \left(a + \frac{b}{x^n}\right) = (a + b x^n) + \left(a + \frac{b}{x^n}\right) \]

Expanding the left-hand side:

\[ a^2 + ab x^n + \frac{ab}{x^n} + b^2 = a + b x^n + a + \frac{b}{x^n} \]

This simplifies to:

\[ a^2 + b^2 + ab(x^n + \frac{1}{x^n}) = 2a + b(x^n + \frac{1}{x^n}) \]

### Step 4: Equate coefficients

From the above equation, we can equate coefficients:

1. For the constant term: \( a^2 + b^2 = 2a \)
2. For the \( x^n \) terms: \( ab = b \)

### Step 5: Solve the equations

From \( ab = b \), we have two cases:

- Case 1: \( b = 0 \) → This leads to \( f(x) = a \), which is constant and does not satisfy \( f(2) = 9 \).
- Case 2: \( a = 1 \) (from \( a^2 - 2a + b^2 = 0 \)) and \( b = 1 \).

This leads us to:

\[ f(x) = 1 + x^n \]

### Step 6: Use the condition \( f(2) = 9 \)

Substituting \( x = 2 \):

\[ f(2) = 1 + 2^n = 9 \]

This simplifies to:

\[ 2^n = 8 \implies n = 3 \]

Thus, we have:

\[ f(x) = 1 + x^3 \]

### Step 7: Verify the solution

Now we need to verify that \( f(x) = 1 + x^3 \) satisfies the original condition:

1. Calculate \( f(2) \):

\[ f(2) = 1 + 2^3 = 1 + 8 = 9 \]

2. Check the functional equation:

\[ f(x) \cdot f\left(\frac{1}{x}\right) = (1 + x^3)\left(1 + \frac{1}{x^3}\right) = 1 + x^3 + \frac{1}{x^3} + 1 = 2 + x^3 + \frac{1}{x^3} \]

And:

\[ f(x) + f\left(\frac{1}{x}\right) = (1 + x^3) + \left(1 + \frac{1}{x^3}\right) = 2 + x^3 + \frac{1}{x^3} \]

Both sides are equal, confirming that our function is correct.

### Final Answer

Thus, the polynomial function is:

\[ f(x) = 1 + x^3 \]