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 $R = \{ f(x) : x \in D\}$ or $R = \{ y : y \in f(D) \}$.

 

[FactChecker]

One can write either $R = \{ f(x) : x \in D\}$ or $R = \{ y : y \in f(D) \}$.

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 $f(x)$.