How do you know a graph represents a function?

If you have a function and you need to verify that a certain graph is its counterpart, then test some points. The first ones should be the simplest. If zero is defined in the function, evaluate the y value at x=0. Look for other values of x that simplify easily and test them. Compare these values with the y values on the graph.

Most functions have predicable shapes just practice more. Some times certain functions will have more efficient ways to be graphed or checked graphically.

If you are given a graph and must make a function then do the opposite of what I have suggested. Look for points (if any) that lie on exact coordinate points. evaluate the rates of change on the graph. look for patterns and build the function the best that you can with your current knowledge.