Relations and Functions June 21, 2012

What is a Relation? A relation is a set of ordered pairs. When you group two or more points in a set, it is referred to as a relation. When you want to show that a set of points is a relation you list the points in braces. For example, if I want to show that the points (-3,1) ; (0, 2) ; (3, 3) ; & (6, 4) are a relation, it would be written like this: {(-3,1) ; (0, 2) ; (3, 3) ; (6, 4)}

Domain and Range Each ordered pair has two parts, an x-value and a y-value. The x-values of a given relation are called the Domain. The y-values of the relation are called the Range. When you list the domain and range of relation, you place each the domain and the range in a separate set of braces.

For Example, {(-3,1) ; (0, 2) ; (3, 3) ; (6, 4)} 1. List the domain and the range of the relation {(-3,1) ; (0, 2) ; (3, 3) ; (6, 4)} Domain: { -3, 0, 3, 6} Range: {1, 2, 3, 4} 2. List the domain and the range of the relation {(-3,3) ; (0, 2) ; (3, 3) ; (6, 4) ; ( 7, 7)} Domain: {-3, 0, 3, 6, 7} Range: {3, 2, 4, 7} Notice! Even though the number 3 is listed twice in the relation, you only note the number once when you list the domain or range!

A function is a relation that assigns each y-value only one x-value. What is a Function? A function is a relation that assigns each y-value only one x-value. What does that mean? It means, in order for the relation to be considered a function, there cannot be any repeated values in the domain. There are two ways to see if a relation is a function: Vertical Line Test Mappings

Using the Vertical Line Test Use the vertical line test to check if the relation is a function only if the relation is already graphed. Hold a straightedge (pen, ruler, etc) vertical to your graph. Drag the straightedge from left to right on the graph. 3. If the straightedge intersects the graph once in each spot , then it is a function. If the straightedge intersects the graph more than once in any spot, it is not a function. A function!

Examples of the Vertical Line Test function Not a function Not a function function ……….

It’s easier than it sounds  Mappings If the relation is not graphed, it is easier to use what is called a mapping. When you are creating a mapping of a relation, you draw two ovals. In one oval, list all the domain values. In the other oval, list all the range values. Draw a line connecting the pairs of domain and range values. If any domain value ‘maps’ to two different range values, the relation is not a function. It’s easier than it sounds 

Example of a Mapping Create a mapping of the following relation and state whether or not it is a function. {(-3,1) ; (0, 2) ; (3, 3) ; (6, 4)} Steps Draw ovals List domain List range Draw lines to connect -3 3 6 1 2 3 4 This relation is a function because each x-value maps to only one y-value.

It is still a function if two x-values go to the same y-value. Another Mapping Create a mapping of the following relation and state whether or not it is a function. {(-1,2) ; (1, 2) ; (5, 3) ; (6, 8)} -1 1 5 6 Notice that even though there are two 2’s in the range, you only list the 2 once. 2 3 8 This relation is a function because each x-value maps to only one y-value. It is still a function if two x-values go to the same y-value.

It is NOT a function if one x-value go to two different y-values. Last Mapping Create a mapping of the following relation and state whether or not it is a function. {(-4,-1) ; (-4, 0) ; (5, 1) ; (3, 9)} Make sure to list the (-4) only once! -1 1 9 -4 5 3 This relation is NOT a function because the (-4) maps to the (-1) & the (0). It is NOT a function if one x-value go to two different y-values. ……….

Vocabulary Review Relation: a set of ordered pairs. Domain: the x-values in the relation. Range: the y-values in the relation. Function: a relation where each x-value is assigned (maps to) on one y-value. Vertical Line Test: using a vertical straightedge to see if the relation is a function. Mapping: a diagram used to see if the relation is a function. ……….

Complete the following questions: 1. Identify the domain and range tell whether if the relation is a function or not: a. {(-4,-1) ; (-2, 2) ; (3, 1) ; (4, 2)} b. {(0,-6) ; (1, 2) ; (7, -4) ; (1, 4)} 2. Use Vertical line test to tell whether if the relation is a function or not: 3. Use a mapping to see if the following relations are functions: a. {(-4,-1) ; (-2, 2) ; (3, 1) ; (4, 2)}

Answers (you will need to hit the spacebar to pull up the next slide) 1a. Domain: {-4, -2, 3, 4} Range: {-1, 2, 1} 1b. Domain: {0, 1, 7} Range: {-6, 2, -4, 4} 2a. 2b. 3a. 3b. Function Not a Function 1 7 -6 2 -4 4 -4 -2 3 4 -1 2 1 Function Not a Function

Functions or Relations? B={(3, -1), (3, -2), (1, -3), (0, -4)} C={(-1, 2), (-2, 1), (-3, 0), (-4, -1)} 4. 5. 6. x -3 -1 1 2 5 6 y 7 9 10 11 12 17 16 x 4 16 36 49 100 y ± 2 ± 4 ± 6 ± 7 ± 10 x ± 3 ± 4 ± 5 ± 6 ± 8 y 10 17 16 37 65