I'm studying real analysis, and I just want to make sure I am understanding the notation of the question correctly:
If $f:\mathbb{R}→\mathbb{R}, x↦x^2,$ prove that...
So am I correct in saying:
domain: {$x \in \mathbb{R}$}
range: {$y \in \mathbb{R}:y=x^2$}
Thanks.
You are not using set notation properly. Usually, the set notation would be $$\{ \text{where the elements are taken from} \mid \text{what condition they must satisfy}\}.$$ The condition must be a well-formed formula with no variables that are not explained.
Neither of the ways you write it meets this standard.
If you want to say that the domain all real numbers, you can say “$\mathrm{domain}(f)=\mathbb{R}$”, or you can say the domain is the set $\{x\mid x\in\mathbb{R}\}$.
For the range, you could say: $$\{ y\in\mathbb{R}\mid \text{there exists a real number }x\text{ such that }y=x^2\},$$ but you can’t just write $\{y\in\mathbb{R}\mid y=x^2\}$, because that condition is incomplete: it does not specify what $x$ is or where it is taken from. But in any case, there is a much better way to describe the range of this function that “all real numbers that are equal to the square of a real number”: you can just say they are the nonnegative real numbers.