Question

The domain of the function \( f(x) = \sqrt{x - 3 + 4\sqrt{5 - x}} \) is

• A. \( [1,2] \)
• B. \( [2,4] \)
• C. \( [3,5] \)
• D. \( [3,20] \)
• E. \( [12,20] \)

Hint:

For multiple square roots, take intersection of all conditions.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For functions involving square roots, the expression inside the square root (the radicand) must be non-negative.
Step 2: Key Formula or Approach:
The function \(f(x)\) is a sum of two terms, each containing a square root. For \(f(x)\) to be defined, both terms must be defined simultaneously. 1. For the term \(\sqrt{x-3}\) to be defined, we must have \(x-3 \ge 0\). 2. For the term \(\sqrt{5-x}\) to be defined, we must have \(5-x \ge 0\). The domain of \(f(x)\) is the intersection of the solutions to these two inequalities.
Step 3: Detailed Explanation:
Condition 1: From the first term, \(\sqrt{x-3}\). The radicand must be non-negative: \[ x - 3 \ge 0 \] \[ x \ge 3 \] In interval notation, this is \([3, \infty)\). Condition 2: From the second term, \(4\sqrt{5-x}\). The radicand must be non-negative: \[ 5 - x \ge 0 \] \[ 5 \ge x \quad \text{or} \quad x \le 5 \] In interval notation, this is \((-\infty, 5]\). Finding the Domain of f(x): The domain of the entire function is the intersection of the domains of its parts. We need to find the values of x that satisfy both \(x \ge 3\) and \(x \le 5\). \[ 3 \le x \le 5 \] In interval notation, this is \([3, 5]\).
Step 4: Final Answer:
The domain of the function is [3, 5].