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

AI Thread Summary
The discussion focuses on the correct notation for the domain and range of the cosecant function, y = csc x. The original attempt used set notation but contained errors, particularly in defining the domain and range. The corrected domain is expressed as D: {x | x ∈ ℝ, x ≠ nπ, n ∈ ℤ}, while the range should be R: {y | y ∈ ℝ \ (-1, 1)}. Participants emphasize the importance of using appropriate symbols and clarifying whether the notation refers to inputs or outputs. The conversation highlights the need for precision in mathematical notation for clarity and correctness.
mcastillo356
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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)\}$$

csc.jpg

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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.)
 
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mcastillo356 said:
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)\}##
 
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Mark44 said:
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##.
 
FactChecker said:
##\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)\}##
 
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FactChecker said:
##\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) \}.
 
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pasmith said:
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)##.
 
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