How do I find the domain of `f(x)=2x`?

Since the problem is very simple, I would like not to bother you with theory or definitions, but just try to think about what your problem means.

When people need explanations about functions, I like to tell them to think of functions as some kind of robot, which takes a number as an input, and gives you a number as an output. The problem is that this operation is not always possible. Take, as a simple example, the function `\frac{1}{x}`. What does this "robot" do? You give him a number, and he gives you back 1 divided by that number. You give him 1, he gives you 1; you give him 2, he gives you `1/2`, and so on.

The problem is that you can't give that robot 0 as an input, since he won't be able to give you back `\frac{1}{0}` as an output.

So, the problem with domains is: "which numbers would my robot refuse?" "When the operation I'm trying to do is not allowed?"

In your case, your function takes a number, and doubles it. Is there any number you can't double? Of course not, since the double of a number is well defined for every number.

When there are no restrictions, there are no values to exclude, and thus your domain is given by the whole real number set, `\mathbb{R}`.