Question

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

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

Hint:

You can test the boundary points of the options to find the answer immediately! Notice what happens if you pick an easy value outside the intervals, like $x = 0$ (present in options C and D): $f(0) = \sqrt{-1} + \sqrt{6}$, which contains an imaginary number. This rules out (C) and (D) instantly. Testing a large value like $x = 7$ rules out (A), leaving (B) as the only mathematically sound choice.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
Determine the domain of a function by testing boundary values from the given options to quickly eliminate invalid intervals.

Step 2: Key Formula or Approach:
The domain consists of all real $x$ for which the function produces real, defined outputs. Substitute strategic test values from each option's interval boundaries and check for undefined or imaginary results.

Step 3: Detailed Explanation:
Testing $x = 0$ (present in options C and D): $f(0) = \sqrt{-1} + \sqrt{6}$, which contains an imaginary component, ruling out both (C) and (D). Next, testing a larger value like $x = 7$ produces an invalid output under the square root constraints, eliminating option (A). Only option (B) survives both checks, producing real values across its entire stated interval.

Step 4: Final Answer:
The correct domain is given by option (B).