- A function describes the relationship between an input variable (x) and an output variable (y).
- A table provides a list of x values and their y values.
- A table is a function if a given x value has only one y value. Multiple x values can have the same y value, but a given x value can only have one specific y value.
How Tables are Created from Functions
Tables are created by inputting numbers into functions. Let’s use the function y = 2x + 1 as an example. This function tells us the relationship between the input variable (x) and the output variable (y). If we input a value for x, we get a specific value for y. The table above shows x values and their y values. An x value of 1 gives us 2*1 + 1. Therefore, if x=1, y=3. An x value of 2 gives us 2*2 + 1. Therefore, if x=2, y=5. And so on.
Each column represents a group of numbers called a “set.” Sets of numbers are presented using { }. The set of x values is called the domain, while the set of y values produced by the function is called the range. In our example, the domain is {1, 2, 3, 4, 5}. The range is {3, 5, 7, 9, 11}. A set of possible y values is known as the “codomain.” This is different from the range, because the range is the actual set of y values. For our example, you can say that the codomain is {1, 3, 5, 7, 9, 11, 13, 15, 17} because these are possible values for y. Meanwhile, the range is {3, 5, 7, 9, 11} because these are the actual values in our table.
How to Figure Out if a Table is a Function
A table is a function if each x value has a specific y value. In our previous example, we created a table using the function y = 2x + 1. But what if we only have a table of numbers? How can we tell that that table came from a function? To figure this out, we’ll need to compare the x values (domain) with the y values (range). Let’s look at some examples.
Example 1 Compare the x values with the y values in the table above. Remember that each x value can only have one possible y value. Is this table a function? Yes. In this table, no x value has more than one possible y value. Therefore, this table is a function.
Example 2 Look closely at the table above. Remember that each x value can only have one possible y value. Is this table a function? Yes. An x value of 3 appears twice, but it always has a y value of 11. This table is a function.
Example 3 Think carefully this time. Does each x value have a specific y value? Yes. The x value of 9 always has a y value of 5 and never something else. Likewise, the x value of 13 always has a y value of 5 and never something else. This is fine because multiple x values can have the same y value, but a given x value can only have one specific y value. Therefore, this table is a function.
How to Tell when a Table is NOT a Function
A table is not a function if a specific x value has more than one y value. For example, let’s say we have a table where an x value of 11 appears twice. The first x value of 11 has a y value of 20. The second x value of 11 has a y value of 16. In this case, the table is not a function because an x value of 11 has more than one y value. Let’s look at some examples in the tables below.
Example 1 In this case, we see that an x value of 3 has a few different y values: 3, 6 and 7. Likewise, an x value of 4 has y values of 2, 19, and 0. This table is not a function since a specific x value can only have one possible y value.
Example 2 You can see that an x value of 5 has two y values: 6 and 11. Since a given x value can only have one specific y value, this table is not a function.
Example 3 Note how an x value of 10 always has a y value of 3, but an x value of 20 has two different y values: 8 and 14. Remember: a specific x value can only have one possible y value. Since 20 has more than one possible y value, it breaks this rule. This means that the table is not a function.
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