Question

The domain of the real valued function \( f(x) = \sqrt{2 + x} + \sqrt{3 - x} \) is:

• A. \( (-2, 3) \)
• B. \( [-2, 3] \)
• C. \( (-2, 3] \)
• D. \( [-2, 3] \)

Hint:

When finding the domain of functions with multiple square roots, determine the valid range for each square root separately and find their intersection.

Answer 1

The Correct Option is D

Step 1: Analyze the function under the square roots. 
- For \( \sqrt{2 + x} \), \( 2 + x \) must be non-negative. 
- For \( \sqrt{3 - x} \), \( 3 - x \) must be non-negative. 
Step 2: Solve the inequalities. 
- \( 2 + x \geq 0 \) implies \( x \geq -2 \). 
- \( 3 - x \geq 0 \) implies \( x \leq 3 \). 
Step 3: Determine the intersection of these intervals to find the domain. 
- The intersection is \( x \in [-2, 3] \).