Find the Domain f(x) = square root of x^2-16

Problem

ƒ(x)=√(,x2−16)

Solution

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

x2−16≥0

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

(x−4)*(x+4)≥0

3. Determine the critical points by setting each factor equal to zero, which gives x=4 and x=−4

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

• For x=−5 (−5)2−16=9≥0 (True)

• For x=0 (0)2−16=−16≥0 (False)

• For x=5 (5)2−16=9≥0 (True)

5. Express the domain in interval notation based on the intervals that satisfy the inequality.

Domain=(−∞,−4]∪[4,∞)

Final Answer

Domain of *ƒ(x)=√(,x2−16)* is *(−∞,−4]∪[4,∞)

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