Functions
Have you ever wondered why you can’t save two computer files with the same name? What if you could? How would you or the computer know which file was which?
What is a function?
A function in math is a relationship between two sets of values where every input value has exactly one output value. What is an input or output value? An input value is the value that goes into the function. An output value is the value created by the input and the function.
There are lots of examples of functions to describe but instead of focusing on more word descriptions let’s look at functions and non-functions (relations), as a GRAPH, TABLE and DATA SET. Notice that anytime an input appears more than once with a different output it is highlighted in yellow.
Picture1.pdfWhat is a relation (or not a function)?
A relation in math is a relationship between two sets of values. More simply put, it is a collection of ordered pairs. Anytime a comparison is being made between two things, you are dealing with a relation. When each input of a relation has only one output, it can be called a function. Here are some famous functions and relations you will more than likely study at some point:
famousfunctionsrelations.pdfHow do you identify a function vs. a relation?
There are two main ways to identify if a relation is a function or just a relation:
1) Vertical Line Test
2) Mapping
Vertical Line Test
Since the inputs (values on the x-axis) of a function can only have one output, the vertical line test is a great way to test a graph to see if it is a function. To use the vertical line test, you draw a vertical line through input values and check to make sure the vertical line only crosses through (intersects) the graph once. If any vertical line crosses through the graph more than one time that graph cannot be classified as a function. Here is the vertical line test in action with the famous functions and relations from the section above:
verticallinetest.pdfMapping
When a graph is not present, another way to test if a relation is a function is by mapping the ordered pairs of the relation. Remember an ordered pair is a set of two coordinates that can be graphed on the coordinate plane. To use mapping, simply draw an arrow from each input value to its output value(s). A function cannot have any input value with more than one arrow. Here is mapping in action using data points from the famous functions and relations:
mapping.pdfWhy do we learn about functions?
There are many different relationships to study in math, and it is important that we take the time to learn about what makes different types of relationships unique. All relationships have been discovered and explored to explain some phenomenon experienced in our world. Learning about functions is one way to start this exploration and classify a specific type of relationship. So many things in our everyday life are based on functions, following the idea that for each input value there is exactly one output value. If you have ever typed on a keyboard, then you have used math outside of the classroom, specifically the math concept of FUNCTIONS. Each key (input) has one output and the keyboard behaves like a mathematical function.
What’s next? To name functions check out Function Notation!
