What's the proper mathematical notation to say Range of Y in Domain of X

Question

How do I correctly express in mathematical notation with no words the bold part below

For F(x) F = 2x when x ranges from 0 to 2, y ranges from 0 to 4

or For for log functions when x ranges from 0 to 1, y ranges from -inf to 0

Answer 1

For the first one you write, $$F([0,2])=[0,4]$$

For the second one your write, $$ log((0,1))=(-\infty, 0)$$

Answer 2

You can simply say:

$$ \begin{aligned} F(x) = 2x \qquad 0 \leq x \leq 2 \\ \therefore 0 \leq F(x) \leq 4 \end{aligned} $$

Answer 3

Notice that, given a function $f:X\to Y$, the expressions $f(x),x\in X$ and $f(A),A\subset X$ are both meaningful expressions (in your case, the argument of your functions can be a real number or a set of real numbers). For example, the range of $f$ is defined as $f(X)$.

Thus, your first statement could be written $F\left([0,2]\right)=[0,4]$, your second, $g\left(]0,1]\right)=]-\infty,0]$.