Find the Domain f(x) = square root of x^2-4

Problem

ƒ(x)=√(,x2−4)

Solution

1. Identify the restriction for a square root function, which requires the radicand (the expression inside the square root) to be greater than or equal to zero.

x2−4≥0

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

(x−2)*(x+2)≥0

3. Determine the critical points by setting each factor to zero, which gives x=2 and x=−2

4. Test the intervals created by the critical points: (−∞,−2] [−2,2] and [2,∞)

5. Evaluate the sign of the product in each interval. For x=−3 (−5)*(−1)=5≥0 (True). For x=0 (−2)*(2)=−4≥0 (False). For x=3 (1)*(5)=5≥0 (True).

6. Combine the intervals where the inequality holds true to define the domain.

(−∞,−2]∪[2,∞)

Final Answer

Domain:(−∞,−2]∪[2,∞)

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