Injective, exhaustive and bijective functions

In the graph of a function we can observe certain characteristics of the functions that give us information about its behaviour.

Observe the graphs of the functions $f(x)=x^2$ and $g(x)=2x$

For the function $f$, we observe that we can trace at least one horizontal straight line ($y$ = constant) that the cuts the graph in more than one point.

For example, if we consider the horizontal straight line $y=4$, we see that there are two points in the domain of $f$, $x=2$, and $x=-2$, with have the same image $f(x)=4$.

On the other hand, any horizontal straight line traced on the graph of the function $g$ will cut its graph only once. Therefore, there aren't any two elements in the domain of $g$ that have the same image.

A function $f$ is injective if every two different elements in its domain have two different images, that is, if it is satisfied that: $$x_1\ne x_2⟹f(x_1)\ne f(x_2)$$

Therefore we observe that the function $g$ is injective while $f$ is not.

If we consider again the functions $f$ and $g$, we observe that: The graph of the function $f$ lies on the positive side of the $y$-axis, thus $Im(f)=[0,+\mathrm{\infty })$.

On the other hand, the graph of the function $g$ is on all the real numbers, thus $Im(g)=\mathbb{R}$.

A function $f$ is exhaustive if its graph coincides with the set of the real numbers, that is, if we have that:$$Im(f)=\mathbb{R}$$

We have therefore that the function $f$ is not exhaustive and that the function $g$ is exhaustive.

Finally

A function is bijective if it is injective and exhaustive simultaneously.

This way the function $g$ of the example is bijective while the function $f$ is not.