Any of the equations from your previous question (function equals its inverse) would also answer the first part of your question. The key to a graph being a function is the vertical line test. If it also passes the horizontal line test, then it's inverse is also a function. So, f(x) = x is a function, and its inverse is also a function. You can easily verify this graphically. f(x) = x2 would be an example of a function whose inverse is a non-function, since the graph of a parabola passes the vertical line test but fails the horizontal line test. So, the second part of your question looks for a sketch of a curve that fails both the vertical and horizontal line tests. A circle would be a simple example, with equation x2 + y2 = 1. If you don't need the equation, you're welcome to sketch any wiggly curve you want that doubles back enough to fail both the vertical and horizontal line tests.
If vertical/horizontal line tests still confuse you, you really ought to set up an online lesson with somebody so we can draw these examples on a whiteboard for you.