Find the Domain square root of x^2-36

Problem

√(,x2−36)

Solution

1. Identify the condition for the square root function to be defined. The radicand (the expression inside the square root) must be greater than or equal to zero.

Domain: *x2−36≥0

2. Factor the difference of squares on the left side of the inequality.

(x−6)*(x+6)≥0

3. Determine the critical points by setting each factor to zero.

x−6=0⇒x=6

x+6=0⇒x=−6

4. Test the intervals created by the critical points: (−∞,−6] [−6,6] and [6,∞)
   For x=−7 (−7)2−36=13≥0 (True)
   For x=0 (0)2−36=−36≥0 (False)
   For x=7 (7)2−36=13≥0 (True)

5. Combine the intervals where the inequality holds true to express the domain.

(−∞,−6]∪[6,∞)

Final Answer

Domain: *(−∞,−6]∪[6,∞)

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