Question

If \( f(x) = \sqrt{x - 3} + 4 \sqrt{5 - x} \), find the domain of \( f(x) \).

• A. \( [3, 5] \)
• B. \( [3, 5) \)
• C. \( (3, 5] \)
• D. \( (3, 5) \)

Hint:

When dealing with square roots, always ensure the expression inside the root is non-negative. Solve the inequalities for each square root separately and combine the results.

Answer 1

The Correct Option is A

We are given the function: \[ f(x) = \sqrt{x - 3} + 4 \sqrt{5 - x} \] To find the domain, we need to ensure that the expressions inside the square roots are non-negative, as the square root of a negative number is undefined for real numbers.

1. Step 1: Find the domain of the first square root \( \sqrt{x - 3} \): For \( \sqrt{x - 3} \) to be defined, we must have: \[ x - 3 \geq 0 \quad \Rightarrow \quad x \geq 3 \] Therefore, the first condition is \( x \geq 3 \).

2. Step 2: Find the domain of the second square root \( \sqrt{5 - x} \): For \( \sqrt{5 - x} \) to be defined, we must have: \[ 5 - x \geq 0 \quad \Rightarrow \quad x \leq 5 \] Therefore, the second condition is \( x \leq 5 \).

3. Step 3: Combine the conditions: The domain of \( f(x) \) is the intersection of the conditions: \[ x \geq 3 \quad \text{and} \quad x \leq 5 \] Thus, the domain of \( f(x) \) is \( [3, 5] \).