Find the Domain complement (A)

Problem

A=√(,x2−9)

Solution

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

x2−9≥0

2. Factor the quadratic expression using the difference of squares formula.

(x−3)*(x+3)≥0

3. Determine the critical points by setting each factor to zero, which gives x=3 and x=−3 These points divide the number line into intervals: (−∞,−3] [−3,3] and [3,∞)

4. Test the intervals to find where the inequality holds true. The inequality is satisfied when x≤−3 or x≥3 This set represents the domain of the function.

Domain=(−∞,−3]∪[3,∞)

5. Find the complement of the domain. The complement consists of all real numbers that are not in the domain. This corresponds to the values where the radicand is negative.

x2−9<0

6. Solve the inequality for the complement. This occurs between the critical points.

−3<x<3

Final Answer

Domain complement=(−3,3)

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