Find the Domain f(x) = square root of x^4-16x^2

Problem

ƒ(x)=√(,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=−5 (−5)^2*((−5)^2−16)=25*(9)=225≥0 (True)

• For x=−1 (−1)^2*((−1)^2−16)=1*(−15)=−15≥0 (False)

• For x=1 (1)^2*((1)^2−16)=1*(−15)=−15≥0 (False)

• For x=5 (5)^2*((5)^2−16)=25*(9)=225≥0 (True)

• For x=0 0^2*(0^2−16)=0≥0 (True)

6. Combine the intervals where the inequality holds true, noting that x=0 is an isolated point that satisfies the inequality.

(−∞,−4]∪{0}∪[4,∞)

Final Answer

Domain:(−∞,−4]∪{0}∪[4,∞)

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