Question

What is the domain and range of `y = tan(2x)`?

Answer 1

Domain: `{x|x!=pi/4+pi/2k,k inZZ}`
Range: `y in RR`

Explanation:

Let us first look at the graph of `y=tan(2x)`:
graph{tan(2x) [-8.41, 9.37, -3.74, 5.146]}
We can see that it has recurring vertical asymptotes, which means that the function is undefined at all these points.

To find the asymptotes, we will look at the following identity:
`tan(theta)=sin(theta)/cos(theta)`
`tan(2x)=sin(2x)/cos(2x)`
This equation tells us that the vertical asymptotes occur when `cos(2x)=0`, and this happens when `x=pi/4+pi/2k` where `k inZZ`

Since the function is defined for all but those `x` values, we just need to exclude them from our domain, so we have:
`{x|x!=pi/4+pi/2k,k in ZZ}`

Next we want to look at the range, and we see on the graph that it goes from `-oo` to `oo` (since each "section" is bounded by two vertical asymptotes), and since the function is continuous on that interval, we know that the domain is all real numbers:
`{y inRR}`