Graphical Representation of the Function
Let's take rectangular Descartes coordinate system; let's draw on coordinate plane all points with x-coordinate x=a, we will obtain line, that is parallel to the x-axis; we say that x=a is equation of this line, in particular, x=0 is equation of y-axis. Similarly, let's draw on coordinate plane all points with y-coordinate y=b, we will obtain line, that is parallel to the x-axis; it is said that y=b is equation of this line, in particular y=0 is equation of x-axis.
Subset F of points of coordinate plane is graph of some function, if it has no more than one common point with any line, parallel to the y-axis.
From this we have following rule to determine whether graph is a graph of some function.
Vertical Line Test. If any vertical line intersects graph at no more than one point, then such graph is graph of function (see figure).
If we are given subset F, that is graph of some function, then it is said, that function is represented graphically. Domain of such function is projection D of set F on x-axis. If we take point `x in D`, then to find corresponding value of the function, we need through point x draw line, parallel to y-axis, line will intersect graph at some point M. y-coordinate of point M is value of function at point x.
For example, consider next graph. This graph is graph of a function. To find value of function at point x=4 we draw vertical line (red) through point x=4. Line intersects graph at point M=(4,3). Therefore, value of function at point x=4 is 3.