Question

How do you graph, identify the domain, range, and asymptotes for `y=-2csc2x-1`?

Answer 1

too long..

Explanation:

I won't use the whole function but a similar one, you can explore urs by lowering it 1 step and the negative version.

Find the domain by checking what `x`'s `y` cannot take on.

As `csc2x=1/{sin(2x)}`, `y` is not defined when `sin2x=0`. So the domain is all `x` except `n\pi/2, n in Z`.

Similary the range is what `y` values is possible for the function to output. Notice how `-1<=sin2x<=1`, therefore `1/{sin2x}` can never reach values between `-1, 1` such as `1/2` or `-1/2`.
So, `y in (-\infty, infty)∖(-1,1)`

The asymptotes is also closely related to the domain, when `y` is divided by `0` vertical asymptotes appear. As earlier this is when `x=n\pi/2, n in Z`

Horisontal asymptotes does not exists as `lim_{x->+-infty} y(x)` does not exist.

How would you graph this?
We aquired a lot of information of `y` so far, so you could start sketching the asymptotes, and by noticing its periodicity. To continue you should plug in known values for `csc2x`.