This may pose as a very obvious question, but I'm kind of new to set theory and don't really know all too much. Suppose you have the function $y=x^2+3$. Let the set of all numbers which $y$ could be (the range) be $R$. Therefore, $y$ is a real number that is greater than or equal to $3$. So would it be accurate to write this set as
$$R=\{y∈\mathbb{R}:y≥3\}$$
I'm not sure if a comma should be used in place of a colon here. Thanks.
Best Answer
Yes, it is fine. In fact, you have solved for the range of $f$ explicitly.
Now, suppose that we do not know how to solve for the range, we can then leave it as
$$\{x^2+3: x \in \mathbb{R}\}$$
If the domain is not $\mathbb{R}$, we can just change the domain in the set notation easily, for example, if the domain is $\mathbb{N}$, then we can just write
$$\{x^2+3: x \in \mathbb{N}\}.$$