Find the domain and range of the function
`f(x) = sqrt(2 - x) + sqrt(1 + x)`

Answer

To find the domain and range of the function \( f(x) = \sqrt{2 - x} + \sqrt{1 + x} \), we will follow these steps:

### Step 1: Determine the conditions for the square roots
The function contains two square root terms: \( \sqrt{2 - x} \) and \( \sqrt{1 + x} \). For these square roots to be defined, the expressions inside them must be non-negative.

1. For \( \sqrt{2 - x} \):
\[ 2 - x \geq 0 \implies x \leq 2 \]

2. For \( \sqrt{1 + x} \):
\[ 1 + x \geq 0 \implies x \geq -1 \]

### Step 2: Combine the conditions
From the inequalities derived, we have:
- \( x \leq 2 \)
- \( x \geq -1 \)

Combining these gives us the domain:
\[ -1 \leq x \leq 2 \]

Thus, the domain of \( f(x) \) is:
\[ \text{Domain} = [-1, 2] \]

### Step 3: Find the range of the function
Next, we will find the range of the function by evaluating it at the endpoints of the domain and checking for any critical points.

1. Evaluate \( f(-1) \):
\[ f(-1) = \sqrt{2 - (-1)} + \sqrt{1 + (-1)} = \sqrt{3} + \sqrt{0} = \sqrt{3} \]

2. Evaluate \( f(2) \):
\[ f(2) = \sqrt{2 - 2} + \sqrt{1 + 2} = \sqrt{0} + \sqrt{3} = \sqrt{3} \]

3. Find the critical points by differentiating \( f(x) \):
\[ f'(x) = \frac{-1}{2\sqrt{2 - x}} + \frac{1}{2\sqrt{1 + x}} \]
Set \( f'(x) = 0 \):
\[ \frac{-1}{2\sqrt{2 - x}} + \frac{1}{2\sqrt{1 + x}} = 0 \implies \frac{1}{\sqrt{1 + x}} = \frac{1}{\sqrt{2 - x}} \]
Squaring both sides:
\[ 1 + x = 2 - x \implies 2x = 1 \implies x = \frac{1}{2} \]

4. Evaluate \( f\left(\frac{1}{2}\right) \):
\[ f\left(\frac{1}{2}\right) = \sqrt{2 - \frac{1}{2}} + \sqrt{1 + \frac{1}{2}} = \sqrt{\frac{3}{2}} + \sqrt{\frac{3}{2}} = 2\sqrt{\frac{3}{2}} = \sqrt{6} \]

### Step 4: Determine the range
From our evaluations:
- Minimum value of \( f(x) \) is \( \sqrt{3} \) (at \( x = -1 \) and \( x = 2 \)).
- Maximum value of \( f(x) \) is \( \sqrt{6} \) (at \( x = \frac{1}{2} \)).

Thus, the range of \( f(x) \) is:
\[ \text{Range} = [\sqrt{3}, \sqrt{6}] \]

### Final Answer
- **Domain**: \( [-1, 2] \)
- **Range**: \( [\sqrt{3}, \sqrt{6}] \)