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A1.F.1.1 Distinguish between relations and functions.

In a Nutshell

Students will understand functions as a well-behaved subdomain of a relation. Relations are simply a mapping from a domain to a range while a function is a type of relationship in which each element of the domain is mapped to only one related element of the range.

Student Actions

Teacher Actions

  • Students are able to identify relations and functions and develop a deep and flexible conceptual understanding that functions are a well-behaved, or well-mapped, subdomain of a relation.
      • Ex: Imagine you’re on the corner of a neighborhood intersection looking for your friend’s house. He says it is the fourth one down on the right. This sets up a relationship to the location and creates a class of houses that might be your friends house… choose any street to go down and the fourth house on the right MIGHT be your friends. Instead, if the directions your friend gave only provided one real option, it would be “well-defined” or “well-behaved”. Perhaps, “go east and my house is the fourth one on the right”. Now we have a mapping that helps us understand not just a general relationship but a very specific relationship. Here we have the initial basis for the difference between a relation and a function. The relation may provide an indication of how to get somewhere but there aren’t enough directions to be confident where to go. A function is an exact map that takes you from point A to B without question.

  • Students Develop mathematical reasoning by exploring all functions as special relations.


    Students will create generalizations about the key characteristics of functions by analyzing various representations of relations and distinguishing between those that only relations and those that are also functions.


  • Provide students with opportunities to use and connect mathematical representations by exploring variables in a relation in order to distinguish between functions and non functions.
  • Pose purposeful questions to encourage students to use the vertical line test as a tool to find functions within a relation plotted on a coordinate plane and not as the definition of a function.

      • Example: “What do you notice about the graph that makes it different than the graphs we have seen before?” “Do we have any tools that can help us determine if this graph is like the others?” “I notice you used the vertical line test (horizontal line test) to suggest this is not (is)  a function, can you justify your reasoning?”

  • Support productive struggle as students work through the difference between a relation and a function, encouraging them to use different representations


Key Understandings


  •  Build an understanding that a function comes from one element in the input (called the domain) being assigned exactly one element from the output  (called the range) ..



Graph A is only a relation since there are 2 ordered pairs with the same x-value and a different y-values at numerous places on the graph.

graph A









Graph B is a relation which is also a function.

graph B

Ordered pairs can be in a table or in a set list.  This table is a relation that is a function, since each x-value is assigned to only one y value f(x).












This table is only a relation and not a function  since the x-value 2 has two different values assigned to it.












  • Students mistake a vertical line for a horizontal  line when investigating the characteristics of the functions.

  • Students don’t realize a function is a special relation.

  • Students interchange the domain values and range values when interpreting a function.




OKMath Framework Introduction

Algebra 1 Introduction




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