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: Number-line view:
The first root needs $x$ at or to the right of 1. The second needs $x$ at or to the left of 6.

Step 2: Overlap:
The overlap of $[1,\infty)$ and $(-\infty,6]$ is $[1,6]$.

Step 3: Test points:
At $x = 0$ the first root has $\sqrt{-1}$, at $x = 7$ the second has $\sqrt{-1}$, at $x = 3$ both are fine.

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