Find the domain and range of `f(x)=sqrt(3-2x-x^2)`

Explanation

To find the domain and range of the function \( f(x) = \sqrt{3 - 2x - x^2} \), we will follow these steps:

### Step 1: Determine the Domain
The function \( f(x) \) is defined only when the expression inside the square root is non-negative. Therefore, we need to solve the inequality:

\[ 3 - 2x - x^2 \geq 0 \]

### Step 2: Rearranging the Inequality
Rearranging the inequality gives:

\[ -x^2 - 2x + 3 \geq 0 \]

Multiplying through by -1 (and reversing the inequality) leads to:

\[ x^2 + 2x - 3 \leq 0 \]

### Step 3: Factor the Quadratic
Next, we factor the quadratic expression:

\[ x^2 + 2x - 3 = (x + 3)(x - 1) \]

Thus, we need to solve:

\[ (x + 3)(x - 1) \leq 0 \]

### Step 4: Find Critical Points
The critical points from the factors are \( x = -3 \) and \( x = 1 \). We will test the intervals defined by these points:

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

### Step 5: Test the Intervals
- For \( x < -3 \) (e.g., \( x = -4 \)):
\((x + 3)(x - 1) = (-1)(-5) > 0\) (not in the solution set)

- For \( -3 < x < 1 \) (e.g., \( x = 0 \)):
\((x + 3)(x - 1) = (3)(-1) < 0\) (in the solution set)

- For \( x > 1 \) (e.g., \( x = 2 \)):
\((x + 3)(x - 1) = (5)(1) > 0\) (not in the solution set)

### Step 6: Include the Endpoints
Since the inequality is less than or equal to zero, we include the endpoints \( x = -3 \) and \( x = 1 \). Thus, the domain is:

\[ [-3, 1] \]

### Step 7: Determine the Range
To find the range, we will analyze the function values at the endpoints and the vertex of the quadratic inside the square root.

1. **Evaluate at the endpoints:**
- \( f(-3) = \sqrt{3 - 2(-3) - (-3)^2} = \sqrt{3 + 6 - 9} = \sqrt{0} = 0 \)
- \( f(1) = \sqrt{3 - 2(1) - (1)^2} = \sqrt{3 - 2 - 1} = \sqrt{0} = 0 \)

2. **Find the vertex:**
The vertex of the quadratic \( -x^2 - 2x + 3 \) occurs at \( x = -\frac{b}{2a} = -\frac{-2}{2(-1)} = -1 \).
- \( f(-1) = \sqrt{3 - 2(-1) - (-1)^2} = \sqrt{3 + 2 - 1} = \sqrt{4} = 2 \)

### Step 8: Conclude the Range
The function takes values from the minimum \( 0 \) to the maximum \( 2 \). Therefore, the range is:

\[ [0, 2] \]

### Final Answer
- **Domain:** \( [-3, 1] \)
- **Range:** \( [0, 2] \)

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