what is the domain of the function f given by f(x)=sqrt x^2-4/x-3

Question

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Answer 1

Hi Aaa from Jeff City.  Used to live in Rolla.  Love that area!

 

I don't know if the (x-3) is under the radical or outside of it, but either way, I believe the domain is the same.

 

To start with, (x-3) in the denominator will give you a problem if it is allowed to become 0 (can't divide by 0), then we know that x CANNOT = 3, or the denominator would become zero and that is not allowed.  

 

That is one restriction on x (the domain is assumed to be all real numbers unless there are restrictions, like in this problem.)  

 

The other restriction comes from the radical.  We MUST KEEP IT POSITIVE, or ZERO.   

 

So set x² - 4 > 0  and find that x² > 4.    When you square root both sides, you get  x  <=>  + 2

 

This is a bit problematic, because it really is both > for the +2, and < for the -2.

 

So the final restrictions are that x must be greater than or equal to 2, but it cannot equal 3.   Or it must be less than or equal to (-2).

 

In interval notation the domain would be:

 

(-infinity,-2]U[2,3)U(3,+infinity)