The domain of the function `sqrt((2-x)(x-3))`

Options

A. `(0,infty)`
B. `(0,infty)`
C. [2,3]
D. -2.3

Answer

To find the domain of the function \( f(x) = \sqrt{(2-x)(x-3)} \), we need to ensure that the expression inside the square root is non-negative. This means we need to solve the inequality:

\[ (2-x)(x-3) \geq 0 \]

### Step 1: Identify the critical points
To find the critical points, we set the expression equal to zero:

\[ (2-x)(x-3) = 0 \]

This gives us two critical points:

1. \( 2 - x = 0 \) → \( x = 2 \)
2. \( x - 3 = 0 \) → \( x = 3 \)

### Step 2: Test intervals around the critical points
The critical points divide the number line into three intervals:

1. \( (-\infty, 2) \)
2. \( [2, 3] \)
3. \( (3, \infty) \)

We will test a point from each interval to determine where the expression is non-negative.

- **Interval 1: \( (-\infty, 2) \)**
Choose \( x = 1 \):
\[ (2-1)(1-3) = 1 \cdot (-2) = -2 \quad (\text{negative}) \]

- **Interval 2: \( [2, 3] \)**
Choose \( x = 2.5 \):
\[ (2-2.5)(2.5-3) = (-0.5)(-0.5) = 0.25 \quad (\text{positive}) \]

- **Interval 3: \( (3, \infty) \)**
Choose \( x = 4 \):
\[ (2-4)(4-3) = (-2)(1) = -2 \quad (\text{negative}) \]

### Step 3: Determine where the expression is non-negative
From our tests, we find:

- The expression is negative in the interval \( (-\infty, 2) \).
- The expression is positive in the interval \( [2, 3] \).
- The expression is negative in the interval \( (3, \infty) \).

### Step 4: Include the endpoints
Since the square root function is defined for zero, we include the endpoints \( x = 2 \) and \( x = 3 \) where the expression equals zero.

### Conclusion
The domain of the function \( f(x) = \sqrt{(2-x)(x-3)} \) is:

\[ \text{Domain: } [2, 3] \]