domain of `f(x)=sqrt(4x -x^2)` is-

Options

A. `R-[0,4]`
B. `R-(0,4)`
C. `(0,4)`
D. `[0,4]`

Answer

To find the domain of the function \( f(x) = \sqrt{4x - x^2} \), we need to determine the values of \( x \) for which the expression under the square root is non-negative. This means we need to solve the inequality:

\[ 4x - x^2 \geq 0 \]

### Step 1: Rearranging the Inequality
First, we can rearrange the inequality:

\[ -x^2 + 4x \geq 0 \]

This can be rewritten as:

\[ x^2 - 4x \leq 0 \]

### Step 2: Factoring the Quadratic
Next, we factor the quadratic expression:

\[ x(x - 4) \leq 0 \]

### Step 3: Finding the Critical Points
The critical points of the inequality occur when the expression equals zero:

\[ x = 0 \quad \text{and} \quad x = 4 \]

### Step 4: Testing Intervals
Now, we will test the intervals determined by these critical points: \( (-\infty, 0) \), \( (0, 4) \), and \( (4, \infty) \).

1. **Interval \( (-\infty, 0) \)**: Choose \( x = -1 \):
\[ (-1)(-1 - 4) = (-1)(-5) = 5 \quad (\text{positive}) \]

2. **Interval \( (0, 4) \)**: Choose \( x = 2 \):
\[ (2)(2 - 4) = (2)(-2) = -4 \quad (\text{negative}) \]

3. **Interval \( (4, \infty) \)**: Choose \( x = 5 \):
\[ (5)(5 - 4) = (5)(1) = 5 \quad (\text{positive}) \]

### Step 5: Determining the Valid Interval
From the tests, we find that the expression \( x(x - 4) \leq 0 \) is satisfied in the interval \( [0, 4] \).

### Conclusion
Thus, the domain of the function \( f(x) = \sqrt{4x - x^2} \) is:

\[ \text{Domain} = [0, 4] \]