If `f(x)-3f((1)/(x))=2x+3(x ne 0)` then f(3) is equal to

Options

A. `-(3)/(2)`
B. `-(5)/(2)`
C. `(7)/(2)`
D. `-1`

Answer

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

### Step 1: Substitute \( x = 3 \) into the equation.
We start by substituting \( x = 3 \) into the given equation.

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

### Step 2: Simplify the equation.
Now, simplify the right side of the equation.

\[ f(3) - 3f\left(\frac{1}{3}\right) = 6 + 3 = 9 \]

This gives us our first equation:

\[ f(3) - 3f\left(\frac{1}{3}\right) = 9 \quad \text{(Equation 1)} \]

### Step 3: Substitute \( x = \frac{1}{3} \) into the equation.
Next, we substitute \( x = \frac{1}{3} \) into the original equation.

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

### Step 4: Simplify the equation.
Now, simplify the right side of this equation.

\[ f\left(\frac{1}{3}\right) - 3f(3) = \frac{2}{3} + 3 = \frac{2}{3} + \frac{9}{3} = \frac{11}{3} \]

This gives us our second equation:

\[ f\left(\frac{1}{3}\right) - 3f(3) = \frac{11}{3} \quad \text{(Equation 2)} \]

### Step 5: Let \( f(3) = x \) and \( f\left(\frac{1}{3}\right) = y \).
Now we can rewrite our equations using \( x \) and \( y \):

1. From Equation 1:
\[ x - 3y = 9 \quad \text{(Equation A)} \]

2. From Equation 2:
\[ y - 3x = \frac{11}{3} \quad \text{(Equation B)} \]

### Step 6: Solve the system of equations.
Now we will solve these two equations simultaneously.

From Equation A:
\[ x = 3y + 9 \]

Substituting \( x \) in Equation B:
\[ y - 3(3y + 9) = \frac{11}{3} \]

Expanding this gives:
\[ y - 9y - 27 = \frac{11}{3} \]
\[ -8y - 27 = \frac{11}{3} \]

### Step 7: Isolate \( y \).
To isolate \( y \), we first convert -27 to a fraction:
\[ -8y = \frac{11}{3} + 27 = \frac{11}{3} + \frac{81}{3} = \frac{92}{3} \]
\[ y = -\frac{92}{24} = -\frac{23}{6} \]

### Step 8: Substitute \( y \) back to find \( x \).
Now substitute \( y \) back into Equation A to find \( x \):
\[ x - 3\left(-\frac{23}{6}\right) = 9 \]
\[ x + \frac{69}{6} = 9 \]
Convert 9 to sixths:
\[ x + \frac{69}{6} = \frac{54}{6} \]
\[ x = \frac{54}{6} - \frac{69}{6} = -\frac{15}{6} = -\frac{5}{2} \]

### Conclusion:
Thus, we find that:
\[ f(3) = x = -\frac{5}{2} \]