A function is a relation between two variables (let's say x and y) in which each value for x produces a single value for y. To put it in scientific terms, the independent variable is x and the dependent variable is y. This means that we can change the value of x to produce different values for y. However, the same input x always gives the same output y.
The relation between the variables is based on a predefined "rule". This rule determines how the input produces the output. For instance, in the function f(x) = x + 5, the rule is that for any value of x, the value of y is five greater. So, if x is 3, y is 8; if x is 10, y is 15; and so on...
You may be asked to identify whether a graph or a set of coordinates (x, y) are a function. If you are given a graph, you can perform what is known as the vertical line test: see if any vertical line would touch the graphed line or set of points more than once. A vertical line has a constant value of x for every value of y, so if a vertical line passes through multiple points on the graph, or passes through the line more than once, the graph does not represent a function. Take a look at this graph:
There is no vertical line that would intersect more than one point on the wavy line, so this is a function. Now check out this graph:
Here, any vertical line to the right of the origin (0, 0) will intersect with two lines. In other words, every value of x will have two possible y-values. Not a function!
If you are given a set of ordered pairs, see if any x-value is repeated. If an x-value appears more than once, it should be paired with the same y-value each time. Otherwise, it is not a function because a single input has multiple outputs. Take a look at this set: (-2, 1), (-1, 3) (0, 5), (1, 7), (2, 9). Each x-value is different, so this is a function.
Now consider this set: (1, 2), (3, 5), (2, 4), (2, 7). Here, the x-value 2 is repeated, with y-values 4 and 7. Since the same input (x = 2) gives two different outputs (y = 4 and y = 7), this set is not a function.