Question

How do you find the domain and range of `g(x)=sqrt(x^2-4)`?

Answer 1

Domain: `x <= -2` or `x>= 2` or `(-oo, 2] uu [2, oo)`
Range: All real numbers or `(-oo, oo)`

Explanation:

Domain refers to all the x-values on the graph.
In a square root function, to get real x-values, we need it to be `0` (`sqrt(0) = 0` or any positive number. Therefore, we will set the square root stuff equal to or more than `0`, like this:
`sqrt(x^2-4) >= 0`
`x^2-4 >= 0`
`x^2 >= 4`
`x <= -2` or `x>= 2` or `(-oo, 2] uu [2, oo)`

Range refers to all the y-values on the graph. Since all the values must be greater than or equal to 0, that means that the range is all real numbers, or `(-oo, oo)`

Here is the graph of this function (there should be arrows at the ends of the graph):