If `2f (x) - 3f (1/ x) = x^2`, x is not equal to zero, then f (2) is equal to

Options

A. `5//2`
B. `-7//4`
C. `-1`
D. None of these

Answer

To solve the equation \( 2f(x) - 3f\left(\frac{1}{x}\right) = x^2 \) for \( f(2) \), we will follow these steps:

### Step 1: Substitute \( x = 2 \) in the equation
We start by substituting \( x = 2 \) into the given equation:

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

This simplifies to:

\[ 2f(2) - 3f\left(\frac{1}{2}\right) = 4 \tag{1} \]

### Step 2: Substitute \( x = \frac{1}{2} \) in the equation
Next, we substitute \( x = \frac{1}{2} \) into the same equation:

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

This simplifies to:

\[ 2f\left(\frac{1}{2}\right) - 3f(2) = \frac{1}{4} \tag{2} \]

### Step 3: Solve the system of equations
Now we have a system of two equations:

1. \( 2f(2) - 3f\left(\frac{1}{2}\right) = 4 \)
2. \( 2f\left(\frac{1}{2}\right) - 3f(2) = \frac{1}{4} \)

Let's solve these equations.

From equation (1), we can express \( f\left(\frac{1}{2}\right) \):

\[ 3f\left(\frac{1}{2}\right) = 2f(2) - 4 \]
\[ f\left(\frac{1}{2}\right) = \frac{2f(2) - 4}{3} \tag{3} \]

Now substitute equation (3) into equation (2):

\[ 2\left(\frac{2f(2) - 4}{3}\right) - 3f(2) = \frac{1}{4} \]

Multiply through by 3 to eliminate the fraction:

\[ 2(2f(2) - 4) - 9f(2) = \frac{3}{4} \]
\[ 4f(2) - 8 - 9f(2) = \frac{3}{4} \]
\[ -5f(2) - 8 = \frac{3}{4} \]

### Step 4: Isolate \( f(2) \)
Now, we isolate \( f(2) \):

\[ -5f(2) = \frac{3}{4} + 8 \]
Convert 8 to a fraction with a denominator of 4:

\[ -5f(2) = \frac{3}{4} + \frac{32}{4} = \frac{35}{4} \]
\[ f(2) = -\frac{35}{4 \cdot 5} = -\frac{35}{20} = -\frac{7}{4} \]

Thus, we find:

\[ f(2) = -\frac{7}{4} \]

### Final Answer
Therefore, \( f(2) \) is equal to \( -\frac{7}{4} \).

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