Identifying Functions from Tables (Examples & Tips)

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Recognizing when a table is (or isn’t) a function is easier than you think
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Cramming for a math test? Struggling with a homework assignment on tables and functions? If this sounds like you, you’re not alone. Tables and functions can be hard to wrap your head around, and understanding how they work can make a big difference in your grade. The tricky part is that some tables are functions and some aren’t. So how do you tell the difference? It may sound like a hard question, but don’t sweat. There are a few dead giveaways that tell you whether a table is a function, and they’re easy to spot if you know where to look. Here’s a step-by-step guide on how to tell when a table is a function—and when it’s not.

Things You Should Know

  • 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.
Section 1 of 3:

How Tables are Created from Functions

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  1. 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). [1] 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.
  2. 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. [2] [3] [4]
    • 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.” [5] 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.
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Section 2 of 3:

How to Figure Out if a Table is a Function

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  1. 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.
  2. 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.
  3. 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.
  4. 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.
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Section 3 of 3:

How to Tell when a Table is NOT a Function

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  1. 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.
  2. 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.
  3. 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.
  4. 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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      References

      1. ↑ https://www.britannica.com/science/function-mathematics
      2. ↑ https://www.mathsisfun.com/sets/domain-range-codomain.html
      3. ↑ https://www.mathsisfun.com/sets/domain-range-codomain.html
      4. ↑ https://www.mathsisfun.com/sets/domain-range-codomain.html
      5. ↑ https://www.mathsisfun.com/sets/domain-range-codomain.html

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