Shortcomings of Function Graphs

Graphs Are Not (Literally) Functions

First of all, graphs are only the visual representation of the function. You can never get the exact form of a function only by looking at a graph.

Look at the following graph:

Actually what I plotted is the graph of $y=1.01x+0.001\sin(x)+0.999$.

As you see, the naked eye cannot get the exact graph—not even computers can. If you take the coordinate of some points, there would still exist some other functions that goes through the same points.

To make things worse, there are graphs that violates the definition of a function.

Here's an example.

As you see, a function is defined only if there's at most one value of $y$ for all $x$ in its domain. This graph clearly cannot be represented by a function.

Nevertheless, graphs are useful in getting a sense how a function should behave, but as we can get a graph from a function, we cannot get a function from a graph. (We might be able to estimate, but not get its exact form.)