Identifying Functions with Ordered Pairs, Tables & Graphs

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  • 0:04 Functions
  • 1:37 Ordered Pairs
  • 2:30 Tables
  • 3:15 Graphs
  • 4:44 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Functions are an extremely important part of mathematics. Let's look at some different ways of representing a function and how to determine if a given representation is, in fact, a function.


Think back to the last time you ate at a restaurant, and try to recall the dessert menu. It probably looked something like this:


See how the price of the dessert is determined by the type of dessert? In other words, if I tell you the type of dessert I want, you can determine the price. This relationship is an example of a function. In a function, one variable is determined by the other. We take an input, plug it into the function, and the function determines the output. For example, if I told you I wanted tapioca pudding, you put it into the dessert menu function, and tell me that it costs $3.

Functions have many representations. In the previous example, we described the function in words, and the image of the dessert menu described the function using a mapping, which related the type of dessert to the price. We can determine that this is a function by making sure that a type of dessert doesn't map to two different prices. If this was the case, we couldn't determine the price from the type of dessert. For example, consider this dessert menu:


If I told you I wanted tapioca pudding, you couldn't tell me if it cost $2 or $3, so this wouldn't represent a function.

In general, a relationship is a function if for every input, there is exactly one output. When this is the case, we can determine the output based on the input. Let's consider a few more representations of functions and how to identify a function from these representations.

Ordered Pairs

A function can be represented using ordered pairs. We simply write the inputs as the first coordinates and the outputs as the second coordinates. Consider our dessert menu example. We would represent this using ordered pairs like this:


By our function rule, no input can have more than one output, so a set of ordered pairs is a function as long as no two ordered pairs have the same first coordinate with different second coordinates. This is illustrated here:


The first set of ordered pairs is a function, because no two ordered pairs have the same first coordinates with different second coordinates. The second example is not a function, because it contains the ordered pairs (1,2) and (1,5). These have the same first coordinate and different second coordinates.


We can use tables to represent functions by listing the input values in one column and the corresponding output values in another column. Let's look at our dessert function using a table.


To determine if a table represents a function, we think back to our rule. No input can have more than one output. Thus, in our table, we can't have two entries with the same input and different outputs. Consider these tables:


The first table represents a function since there are no entries with the same input and different outputs. The second table is not a function, because two entries that have 4 as their input, but one has 7 as the output, and the other has 14 as the output.

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