A relation is basically any ordered pair you come across. In the case of the pair (x,y), when you have x it corresponds to the accompanying pair (y). All relations have more than a single possible output for one input. Functions are a bit stricter; they are ordered pairs that have a fixed relationship where every input only has one output. Because of this all functions are relations, but not all relations are functions. In this batch of worksheets you will need to compare ordered pairs and determine if the relation that you are presented with is a function.

These worksheets we look into relations and functions that meet those requirements. Your students will use these worksheets in order to practice identifying whether a given relation is also a function. Students will also find inverse relations, identify domains and ranges, and more.

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Study the series of points plotted on each of the grids and determine if they represent a function. Three problems are provided.

How to Tell if a Relation is a Function

What is a Relation?

A relation is between the two coordinates that map the input and output. For example, y2=x. In this example, given the input (x), there can be multiple outputs (y). if x=4, then y can either be 2 or -2.

In simple terms, relations are different points on a map that come in ordered pairs. When making a graph, we need both x-coordinated and their corresponding y-coordinates. These two coordinates have a simple connection to one another. Relations can be represented in the form of a table, a graph, or a mapping diagram.

What is a Function?

A function can be categorized as a sub-set of relations. They represents the relation between the two coordinate pairs (x and y). Functions have a domain and range which depict the input and the output values.

The domain is a collection of first values in a pair that shows the set of all input values. The range is a collection of the second values in a pair that show the set of all output values.

For example, in the relation {(2,3), (4,5), (6,7),(8,9)}, the domain values would be {2,4,6,8} and the range values would be {3,5,7,9}.

A function can be defined as a relation showing that every x value is associated with one y value.

When is a Relation a Function?

The only way to know this is to look at the set of ordered pairs and see if they follow the rule that every input value must be associated with an output value.

Relations can be divided into four main types:

One to one: Where each input (x) value corresponds to one output (y) value. One to many: Where one input (x) value corresponds to multiple output (y) values. Many to one: Where multiple input (x) values correspond to the same output (y) value. Many to many: Where multiple input (x) values correspond to multiple output (y) values.

A relation can only be called a function if it is one to one or many to one. So, it is only a function if the x values map to only one y value.

You can easily distinguish if this relationship exists in a table or a mapping diagram, as you will only have to see that one or many x values do not correspond to more than one y value. You can distinguish in a graph using the vertical line test; draw a vertical line on the graph and see how many points it touches. If it only touches one point, the relation is a function.

Understanding the differences between these two is easy once you know precisely what they are. Functions are just a specific type of relations, and once you've understood that, you will never question yourself again on this.

How are Relations and Functions Different?

Understanding the difference between the relations and the functions is not as easy as you think. It is a little bit difficult because they are closely associated with each other. We always need to go in-depth and learn this difference with full consideration.
The Definitional Difference between the two :
Relation - In math, it is the group of ordered pairs that have an object from one set to the other one. For example,
We have two sets X and Y. The object (a) belongs to the set X, and object (b) belongs to the set Y. So, we can say that both objects (a, b) of the two different sets are ordered pairs, and they have relations.
Functions - We can define it by the input set to the set of outputs. According to the function, in the group X of input must have the same and only a single output in the set Y.
Symbols - We express relation by the symbol 'R' and function by the symbol 'F' or 'f.'
Example for R - R = {(2, x), (9,y), (2,z)}
It will not be a function when 2 is the input for X and Z.
Example for F - F = {(2, x), (9,y), (5,x)}. Note - Every relation cannot be a function, but every function is a relation.