Find the Domain square root of x^4-16x^2

Problem

√(,x^4−16*x^2)

Solution

1. Identify the condition for the domain of a square root function, which requires the radicand to be greater than or equal to zero.

x^4−16*x^2≥0

2. Factor the expression by taking out the greatest common factor, x^2

x^2*(x^2−16)≥0

3. Factor the difference of squares inside the parentheses.

x^2*(x−4)*(x+4)≥0

4. Determine the critical points by setting each factor to zero, which gives x=0 x=4 and x=−4

5. Test the intervals created by the critical points: (−∞,−4] [−4,0] [0,4] and [4,∞)

• For x∈(−∞,−4] the expression is positive.

• For x∈[−4,4] the expression is negative or zero (specifically, it is zero at x=−4,0,4.

• For x∈[4,∞) the expression is positive.

6. Combine the intervals where the inequality holds true. Since x^2 is always non-negative, the inequality x^2*(x^2−16)≥0 is satisfied when x^2−16≥0 or when x=0

Final Answer

Domain:(−∞,−4]∪{0}∪[4,∞)

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