Question

The domain of the function \(f(x) = \sqrt{x-1}+\sqrt{6-x}\) is...

• A. \([1,\infty )\)
• B. \([1,6]\)
• C. \((-\infty ,1)\)
• D. \((6,\infty )\)

Hint:

Both expressions under the roots must be non-negative.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A square root is defined only for non-negative arguments. A sum is defined where each term is defined.

Step 2: Conditions:
\(x - 1 \ge 0 \Rightarrow x \ge 1\).
\(6 - x \ge 0 \Rightarrow x \le 6\).

Step 3: Combine:
Both must hold, so \(1 \le x \le 6\). The domain is \([1, 6]\).

Step 4: Why the other options are wrong.
\([1,\infty)\) forgets the second root. \((-\infty,1)\) and \((6,\infty)\) each make one of the radicands negative.

Final Answer:
The domain is \([1, 6]\), option (B). \[ \boxed{[1,6]} \]