Cosecant function: notation for the domain and range -- Am I right?

[mcastillo356]

TL;DR Summary

The net is very confusing; the textbooks don't make it easy. Is it right my attempt?

Hi, PF

There are two ways to write domain and range of a function: through set notation, or showing intervals.

I've chosen the set notation, and, for ##y=\csc x##, this is the attempt:

$$\text{D}:\{x\,|\,x\not\in{n\pi},\,n\not\in{\mathbb{Z}}\}$$
$$\text{R}:\{f(x)\,|\,x=\mathbb{R}\(-1,1)\}$$



PD: Post whitout preview

 

[FactChecker]

You are thinking right. Only your notation has some mistakes.
For ##\mathbf{D}##, try to express it this way: you want all ##x## in ##\mathbb{R}## minus the set {##n\pi |n \in \mathbb{Z}## }
For ##\mathbf{R}##, do you want to restrict the input, ##x##, or the values of the range, ##y##? Instead of ##x = \mathbb{R}## ..., don't you want ##y \in \mathbb{R}## ...? (Of course, you can still use the dummy variable ##x## instead of ##y##, but I think that ##y## is more traditional for some people.)

 

[Mark44]

I've chosen the set notation, and, for ##y=\csc x##, this is the attempt:

$$\text{D}:\{x\,|\,x\not\in{n\pi},\,n\not\in{\mathbb{Z}}\}$$
$$\text{R}:\{f(x)\,|\,x=\mathbb{R}\(-1,1)\}$$

Better:
##\text{D}:\{x\,|\,x \ne n\pi,\,n \in{\mathbb{Z}}\}##
##\text{R}:\{f(x)\,|\,x \in \mathbb{R} \text{\\} (-1,1)\}##

 

[FactChecker]

Better:
##\text{D}:\{x\,|\,x \ne n\pi,\,n \in{\mathbb{Z}}\}##
##\text{R}:\{f(x)\,|\,x \in \mathbb{R} \text{\\} (-1,1)\}##

##\text{R}## is wrong. It shouldn't be ##f(x)##. It should just be ##x##.

 

[Mark44]

##\text{R}## is wrong. It shouldn't be ##f(x)##. It should just be ##x##.

Revised version:
##\text{D}:\{x\,|\,x \in \mathbb R, x \ne n\pi,\,n \in{\mathbb{Z}}\}##
##\text{R}:\{y\,|\,y \in \mathbb{R} \text{\\} (-1,1)\}##

 

[pasmith]

##\text{R}## is wrong. It shouldn't be ##f(x)##. It should just be ##x##.

One can write either [itex]R = \{ f(x) : x \in D\}[/itex] or [itex]R = \{ y : y \in f(D) \}[/itex].

 

[FactChecker]

One can write either [itex]R = \{ f(x) : x \in D\}[/itex] or [itex]R = \{ y : y \in f(D) \}[/itex].

I would consider that a good generic definition of the range, but I think that a homework problem would want an answer that specifically states the elements of the range without simply using the generic symbol ##f(x)##.