What Is The Domain Of Y=1/221ax\Xb2-4 In Set Builder Notation?

What is the domain of y=1/√x²-4 in set builder notation?

 

You want to find the domain of 1x2−4−−−−−√.

We need two things: the denominator must be defined, and nonzero.

The denominator is equal to zero when x=±2.

We also have x2−4<0 if −2<x<2, so the denominator is not defined for x∈(−2,2). *

Therefore, the function is NOT defined on x∈−2,2, and thus the function IS defined on {x∈R|x<−2 or x>2}

Or more simply, {x∈R∣|x|>2}

*Notice that x2−4<0⟹x2<4⟹|x|<2, since x2−−√=|x|.

Then, since |x|=2 also makes f(x) undefined, we have f(x) is defined for reals where |x|>2.