Let `f (x) =(2 |x| -1)/(x-3)`
Range of `f (x):`

Options

A. `R- {3}`
B. `(-oo, (1)/(3) ] uu (2,oo)`
C. `(-2, (1)/(3)] uu (2,oo)`
D. R

Answer

To find the range of the function \( f(x) = \frac{2 |x| - 1}{x - 3} \), we will analyze the function for two cases based on the definition of the absolute value.

### Step 1: Analyze the function for \( x \geq 0 \)

When \( x \geq 0 \), \( |x| = x \). Thus, the function simplifies to:
\[ f(x) = \frac{2x - 1}{x - 3} \]
Let \( f(x) = y \):
\[ y = \frac{2x - 1}{x - 3} \]
Cross-multiplying gives:
\[ y(x - 3) = 2x - 1 \]
Rearranging this, we have:
\[ yx - 2x = 3y - 1 \]
Factoring out \( x \):
\[ x(y - 2) = 3y - 1 \]
Thus, we can express \( x \) as:
\[ x = \frac{3y - 1}{y - 2} \]
Since \( x \geq 0 \), we need:
\[ \frac{3y - 1}{y - 2} \geq 0 \]
This inequality holds when both the numerator and denominator are positive or both are negative.

### Step 2: Determine when \( 3y - 1 \geq 0 \)

1. \( 3y - 1 \geq 0 \) implies:
\[ y \geq \frac{1}{3} \]

### Step 3: Determine when \( y - 2 > 0 \)

2. \( y - 2 > 0 \) implies:
\[ y > 2 \]

### Step 4: Combine the conditions

From the analysis above, we have:
- \( y \geq \frac{1}{3} \)
- \( y > 2 \)

Thus, for \( x \geq 0 \), the range of \( f(x) \) is \( [\frac{1}{3}, 2) \).

### Step 5: Analyze the function for \( x < 0 \)

When \( x < 0 \), \( |x| = -x \). The function becomes:
\[ f(x) = \frac{-2x - 1}{x - 3} \]
Let \( f(x) = y \):
\[ y = \frac{-2x - 1}{x - 3} \]
Cross-multiplying gives:
\[ y(x - 3) = -2x - 1 \]
Rearranging this, we have:
\[ yx + 2x = 3y - 1 \]
Factoring out \( x \):
\[ x(y + 2) = 3y - 1 \]
Thus, we can express \( x \) as:
\[ x = \frac{3y - 1}{y + 2} \]
Since \( x < 0 \), we need:
\[ \frac{3y - 1}{y + 2} < 0 \]
This inequality holds when the numerator and denominator have opposite signs.

### Step 6: Determine when \( 3y - 1 < 0 \)

1. \( 3y - 1 < 0 \) implies:
\[ y < \frac{1}{3} \]

### Step 7: Determine when \( y + 2 > 0 \)

2. \( y + 2 > 0 \) implies:
\[ y > -2 \]

### Step 8: Combine the conditions

From the analysis above, we have:
- \( y < \frac{1}{3} \)
- \( y > -2 \)

Thus, for \( x < 0 \), the range of \( f(x) \) is \( (-2, \frac{1}{3}) \).

### Step 9: Combine both ranges

Combining the ranges from both cases, we get:
\[ \text{Range of } f(x) = (-2, \frac{1}{3}) \cup (2, \infty) \]

### Final Answer

Thus, the range of \( f(x) \) is:
\[ (-2, \frac{1}{3}) \cup (2, \infty) \]