Question

What is the domain and range of `y=csc x`?

Answer 1

Domain: all `x in RR: x != n pi forall n inZZ`
Range: `(-oo, -1] uu [+1, +oo)`

Explanation:

`y = cscx`

`y = 1/sinx`

`sin x` is defined `forall x in RR`

`csc x` is defined wherever `sinx != 0 -> x != npi forall n in ZZ`

Hence, the domain of `y` is all `x in RR: x != n pi forall n inZZ`

Consider: `-1<=sinx<=+1 forall x in RR`

Hence, the local maxima of `y` are `1/-1 = -1`

and the local minima of `y` are `1/1 = 1`

Then, since `y` has no upper or lower bounds its range is:
`(-oo, -1] uu [+1, +oo)`

This maybe more easily visualised by the graph of `cscx` below.

graph{cscx [-8.89, 8.885, -4.444, 4.44]}