Find the Domain f(x) = square root of x^4-16x^2
Problem
Solution
Identify the condition for the domain of a square root function, which requires the radicand to be greater than or equal to zero.
Factor the expression by taking out the greatest common factor,
x2
Factor the difference of squares inside the parentheses.
Determine the critical points by setting each factor to zero, which gives
x=0 x=4 andx=−4 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*(0−16)=0≥0 (True)
Combine the intervals where the inequality holds true, noting that
x=0 is an isolated point that satisfies the inequality.
Final Answer
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