Not all the elements of the domain need to belong to the real numbers.
Consider the function $$f(x)=\displaystyle \sqrt{x3}$$ where we take the positive solution of the square root, only has positive images.
We will only consider those $$x$$ in the domain that belong to the real number, in which case all the real numbers that are greater than or equal to $$0$$.
We will call the image of a function $$f$$ the set of real numbers that are an image of $$f$$ of the elements in its real domain. It will be denoted by $$Im (f)$$.
Therefore the image of the function $$f(x)=\sqrt{x3}$$ is $$Im (f) = [0, +\infty)$$
Find the image of the following functions:
 $$f (x) = 2x  1$$
 $$f(x)=3x^2$$

$$f(x)=\displaystyle \frac{1}{x}$$

The function can take any real number as an image. Therefore, $$Im (f) =\displaystyle \mathbb{R}$$.

In this case the function only has positive images or $$0$$, since the square of a number cannot be a negative. Therefore $$Im (f) = [0, +\infty)$$
 Finally, the function can take any real value except $$0$$, since it $$f(x)=\displaystyle \frac{1}{x}$$ only when $$x=0$$ we have that the image is not defined in $$\mathbb{R}$$.
Therefore, $$Im (f) =\mathbb{R}{0}$$.