Which of the following function x has range R :

Options

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

Answer

To determine which of the given functions has a range of all real numbers (R), we will analyze each function step by step.

### Step 1: Analyze the First Function
**Function:** \( f(x) = \frac{x - 1}{x - 3} \)

1. Set \( f(x) = y \):
\[ y = \frac{x - 1}{x - 3} \]
2. Rearranging gives:
\[ y(x - 3) = x - 1 \implies yx - 3y = x - 1 \implies yx - x = 3y - 1 \implies x(y - 1) = 3y - 1 \]
3. Thus, we have:
\[ x = \frac{3y - 1}{y - 1} \]
4. The function is undefined when \( y - 1 = 0 \) (i.e., \( y = 1 \)). Therefore, \( f(x) \) cannot take the value \( 1 \).

**Conclusion:** The range is \( \mathbb{R} \setminus \{1\} \).

### Step 2: Analyze the Second Function
**Function:** \( g(x) = \frac{x - 3}{(x - 1)(x - 2)} \)

1. Set \( g(x) = y \):
\[ y = \frac{x - 3}{(x - 1)(x - 2)} \]
2. Rearranging gives:
\[ y(x - 1)(x - 2) = x - 3 \]
3. Expanding yields:
\[ y(x^2 - 3x + 2) = x - 3 \implies yx^2 - 3yx + 2y - x + 3 = 0 \]
4. The discriminant \( D \) must be non-negative for \( x \) to be real:
\[ D = (-3y - 1)^2 - 4y(2y + 3) \geq 0 \]
5. Solving the discriminant leads to a quadratic inequality in \( y \).

**Conclusion:** The range does not include all real numbers.

### Step 3: Analyze the Third Function
**Function:** \( h(x) = \frac{(x - 1)(x - 3)}{x - 2} \)

1. Set \( h(x) = y \):
\[ y = \frac{(x - 1)(x - 3)}{x - 2} \]
2. Rearranging gives:
\[ y(x - 2) = (x - 1)(x - 3) \]
3. Expanding yields:
\[ yx - 2y = x^2 - 4x + 3 \implies x^2 - (4 + y)x + (3 + 2y) = 0 \]
4. The discriminant \( D \) must be non-negative:
\[ D = (4 + y)^2 - 4(3 + 2y) \geq 0 \]
5. Solving this gives a quadratic inequality in \( y \).

**Conclusion:** The range includes all real numbers.

### Step 4: Analyze the Fourth Function
**Function:** \( k(x) = \frac{x - 2}{(x - 1)(x - 3)} \)

1. Set \( k(x) = y \):
\[ y = \frac{x - 2}{(x - 1)(x - 3)} \]
2. Rearranging gives:
\[ y(x - 1)(x - 3) = x - 2 \]
3. Expanding yields:
\[ y(x^2 - 4x + 3) = x - 2 \implies yx^2 - 4yx + 3y - x + 2 = 0 \]
4. The discriminant \( D \) must be non-negative:
\[ D = (-4y - 1)^2 - 4y(3y + 2) \geq 0 \]
5. Solving this gives a quadratic inequality in \( y \).

**Conclusion:** The range includes all real numbers.

### Final Results
- **First Function:** Range is \( \mathbb{R} \setminus \{1\} \) (not all real numbers).
- **Second Function:** Range does not include all real numbers.
- **Third Function:** Range is \( \mathbb{R} \) (includes all real numbers).
- **Fourth Function:** Range is \( \mathbb{R} \) (includes all real numbers).

### Answer
The functions with a range of all real numbers are the **Third Function** and the **Fourth Function**.