What is the correct syntax when writing about the domain and range of a function?
For example, let's say:
$f(y)=\cos{y}$ and $y(x)=\arcsin{x}$
In order to simplify the function $f(y(x))=\cos{\arcsin{x}}$ I want to write down the domains and ranges of all functions involved in a thorough way.
So I would start with the domain of $\arcsin{x}$ which is $[-1,1]$ then the range, which is $[-\frac{\pi}{2},\frac{\pi}{2}]$, which is also the domain of $\cos{y}$, which implies that the range of $\cos{y}$ is always positive.
How to write the above using the right mathematical syntax? I mean, using the right symbols?
Further question: what does the "element of" $\epsilon$ symbol mean and how to use it? Is that symbol useful in this case?
Furthermore I also sometimes see symbols like: $\mathfrak{D}$ and $\mathfrak{R}$. How are they correctly used?
Best Answer
There are several mistakes which are probably just the result of sloppy writing, but can also lead to some wrong results:
Other than that, it's hard to give you a full answer because I don't understand your question. Do you want to write down the domain of the function $\cos\arcsin x$? Then that domain is equal to the domain of $\arcsin x$, which is $[-1,1]$. Or do you also want to calculate the range?