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 by considering the two cases for \( |x| \) (i.e., when \( x \geq 0 \) and \( x < 0 \)) and also the behavior of the function around the point where the denominator is zero.

### Step 1: Identify the cases for \( |x| \)
The function has two cases based on the absolute value:
1. Case 1: \( x \geq 0 \) (where \( |x| = x \))
2. Case 2: \( x < 0 \) (where \( |x| = -x \))

### Step 2: Analyze Case 1: \( x \geq 0 \)
For \( x \geq 0 \):
\[ f(x) = \frac{2x - 1}{x - 3} \]
We need to find the behavior of this function as \( x \) approaches different values.

### Step 3: Find limits as \( x \) approaches critical points
- **At \( x = 3 \)**: The function is undefined because the denominator becomes zero.
- **As \( x \) approaches 3 from the left (\( x \to 3^- \))**:
\[ f(x) \to \frac{2(3) - 1}{3 - 3} \to \text{undefined (approaches } +\infty\text{)} \]
- **As \( x \) approaches 3 from the right (\( x \to 3^+ \))**:
\[ f(x) \to \frac{2(3) - 1}{3 - 3} \to \text{undefined (approaches } -\infty\text{)} \]

### Step 4: Find limits as \( x \to \infty \)
- **As \( x \to \infty \)**:
\[ f(x) \to \frac{2x - 1}{x - 3} \approx \frac{2x}{x} = 2 \]

### Step 5: Analyze Case 2: \( x < 0 \)
For \( x < 0 \):
\[ f(x) = \frac{-2x - 1}{x - 3} \]
Again, we need to find the behavior of this function.

### Step 6: Find limits as \( x \) approaches critical points
- **At \( x = 3 \)**: The function is still undefined.
- **As \( x \to -\infty \)**:
\[ f(x) \to \frac{-2(-\infty) - 1}{-\infty - 3} \to \frac{+\infty}{-\infty} \to 0 \]

### Step 7: Combine the results
From our analysis:
- As \( x \) approaches 3 from the left, \( f(x) \to +\infty \).
- As \( x \) approaches 3 from the right, \( f(x) \to -\infty \).
- The function approaches 2 as \( x \) approaches infinity.
- The function approaches -2 as \( x \) approaches negative infinity.

### Step 8: Determine the range
The range of \( f(x) \) can be summarized as:
- From \( -\infty \) to \( -2 \) (not including -2),
- From \( -2 \) to \( \frac{1}{3} \) (including \( \frac{1}{3} \)),
- From \( 2 \) to \( +\infty \) (not including 2).

Thus, the range of \( f(x) \) is:
\[ \text{Range} = (-\infty, -2) \cup \left[-\frac{1}{3}, 2\right) \cup (2, +\infty) \]