A function is a relationship between two variables. One of the variables is the input of the function and the other variable is the output. A key aspect of a function is that every input value produces exactly one output. Now, two different input values can produce the same output value, but a single input value cannot produce two different output values.
The following order pairs, where the first number is the input of a function and the second number is the function’s output.
(0, 3) (1, 4) (2, 7) (3, 8) (4, 12) (5, 12) (6, 8) (7, 7) (8, 4) (9, 3)
The above order pairs indicate that there is a function associated with these order pairs. This is because for every input there is exactly one output. Even though different inputs result in the same outputs, for example, the inputs of 0 and 9 both produce an output of 3.
Take the following order pairs:
(0, 4) (1, 5) (2, 6) (2, 3)
The above order pairs are not associated with a function. This is because with an input of 2 cannot produce an output of 3 and 6 at the same time.
To determine if a relationship is a function graphically you can perform the vertical line test on the function. To do the vertical line test take the following graph.
Then draw vertical lines on the graph, as shown below where the veritcal lines are red. If the vertical lines intersect the graph at atmost one point then the relationship is a function.
Because the vertical lines only intersect the graph at one point in the above graph it is a function.
Take the following graph as another example.
Applying the veritcal line test finds that this is not a function.
This is because the rightmost vertical line crosses the graph at two points. Which means that one input produces two different outputs.