A function is a relationship between input and output. The output or dependent variable depends on the input or independent variable. This relationship ensures that for every input, there is only 1 output.
To better understand this relationship functions can be graphed using a coordinate plane. Our independent variable, or input is our x value. The dependent variable, or output is the y value.
Let's look at the relationship between a car going 60 mph over the course of 5 hours. In this case time cannot change so it is our independent variable. Since the car is going 60 mph, every hour the car's distance traveled will increase by 60 miles.
As can be seen this graph shows an example of a function as for every x-value (input), there is only 1 y-value (output).
The above example is an example of a continuous function. A
Continuous Function is a function that is represented by a smooth line or curve. In this case it is continuous because the car is always moving and has a distance at every point in time.
A function that consits of points not connected is called a
Discrete Function. Let's take a look at a discrete function. Let's say you go to a store that sells ice cream cones. Each cone costs $2.00, however the 5th cone is free! Let's look at the graph. In this problem, the amount of money we have spent on ice cream cones depends on the number of cones we have bought. The number of cones is our independent variable and the amount spend is our dependent variable.
Each time we buy a cone, the amount spent jumps by $2..(except that free one). In this case our data is not connected by a smooth line or curve, so it is a continuous function.
Vertical Line Test
There is a very simple way to tell if a graph is a function or not. Simply draw vertical lines over the function. If any of the lines cross the graph more than once, it is not a function. If each line crosses the graph only once, it is a function.
Here are some examples:
Functions:
Non-Functions:
As shown, our vertical line tests (red lines) show that the first two graphs are functions. The red lines never cross the graph more than once, while on the bottom two graphs, each red line crosses the graph a couple times.
Looking at Data
We can look at data as well to determine if a function exists. The data below describes a function since there is only 1 y-value for every x-value:
| x |
y |
| 1 |
10 |
| 5 |
15 |
| 9 |
8 |
| -8 |
-2 |
| 25 |
10 |
| 14 |
-4 |
| -5 |
0 |
The data below does not describe a function since there are two different y-values for the same x-value:
| x |
y |
| 7 |
10 |
| 3 |
5 |
| 7 |
6 |
| 0 |
-8 |
| -4 |
-2 |
| 1 |
7 |
| 1 |
0 |