Question

What is the range of the function `y = tan x`?

Answer 1

The range of the tangent function is all real numbers! If you look at the graph below:

you will see that the various repeated "pieces" of the graph point both upward and downward in an increasing direction from left to right. The function is undefined at `frac{pi}{2}` and its odd multiples, or `frac{(2n+1)pi}{2}` .
Therefore, infinity and negative infinity describe where this function graph is heading.....that is what we call "all real numbers".