Question

f(x)=√(x^4-16x^2)
Find domain *please tell me how you get the answer*

Answer 1

since you cannot take the square root of a negative number, this means that \[\large x^4-16x^2\ge 0\]


You can factor the LHS to get \[\large x^2(x^2-16)\ge 0\] which factors further to \[\large x^2(x-4)(x+4)\ge 0\]


Solve the last inequality to find the domain

Answer 2

Can you explain it farther I already had it up to that point

Answer 3

since x^2 is ALWAYS nonnegative, this means it has no influence on the answers. So we can ignore it and just focus on solving


\[\large (x-4)(x+4)\ge0\]


From the inequality above, if we plug in either x=-4 or x=4, we'll get 0 on the LHS.


Now let's plug in some test points


If we plug in x=-5 (or some number less than -4), then we get

(-5-4)(-5+4) = -9*-1 = 9 which is greater than zero


So the interval from negative infinity up to -4 (including -4) satisfies the inequality


If we plug in x=0 (or some number in between -4 and 4), then we get

(0-4)(0+4) = -4*4 = -16 which is less than zero


So the interval from -4 to 4 (excluding both) does NOT satisfy the inequality



If we plug in x=5 (or some number greater than 4), then we get

(5-4)(5+4) = 1*9 = 9 which is greater than zero


So the interval from 4 to infinity (including 4) satisfies the inequality


This means that the domain is \[\large \left(-\infty,-4]\cup[4,\infty\right)\]