Relations and Functions

1. Relations and Functions Module 1 Lesson 1

2. 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)}

3. 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 a relation, you place each (the domain and the range) in a separate set of braces.

4. For Example, 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!

5. Finding Domain and Range from a graph What is the domain? The graph to the left is of y = x2, and we can square any number we want. This makes the domainall real numbers. On a graph, the domain corresponds to the horizontal axis.  Since that is the case, we need to look to the left and right to see if there are any end points or holes in the graph to help us find our domain. If the graph infinite to left and to the right, then the domain is represented by all real numbers.

6. If I plug any number into this function, am I ever going to be able to get a negative number as a result? No! The range of this function is all positive numbers which is represented by y ≥ 0. On a graph, the range corresponds to the vertical axis.  We need to look up and down to see if there are any end points or holes to help us find our range. If the graph keeps going up and down with no endpoint then the range is all real numbers. However, this is not the case here. The graph does not ever go below the x-axis, never returning a negative range value. What's the range?

7. 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 ……….

8. 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! Using the Vertical Line Test

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

10. 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 

11. 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 0 3 6 1 2 3 4 This relation is a function because each x-value maps to only one y-value. ……….

12. 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. ……….

13. 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 0 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 goes to two different y-values. ……….

14. Vocabulary Review • Relation: a set of order 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.

15. Practice Complete the following questions and check your answers on the next slide. • Identify the domain and range of the following relations: a.{(-4,-1) ; (-2, 2) ; (3, 1) ; (4, 2)} b. {(0,-6) ; (1, 2) ; (7, -4) ; (1, 4)} • Graph the following relations and use the vertical line test to see if the relation is a function. Connect the pairs in the order given. a. {(-3,-3) ; (0, 6) ; (3, -3)} b. {(0,6) ; (3, 3) ; (0, 0)} • Use a mapping to see if the following relations are functions: a. {(-4,-1) ; (-2, 2) ; (3, 1) ; (4, 2)} b. {(0,-6) ; (1, 2) ; (7, -4) ; (1, 4)}

16. -4 -2 3 4 0 1 7 -1 2 1 -6 2 -4 4 Answers (you will need to hit the spacebar to pull up the next slide) 1a. Domain: {-4, -2, 3, 4} Range: {-2, 2, 1} 1b. Domain: {0, 1, 7} Range: {-6, 2, -4, 4} 2a. 2b. 3a. 3b. Function Not a Function Not a Function Function

17. Function Notation The equation that represents a function is called a function rule. • A function rule is written with two variables, x and y. • It can also be written in function notation using f(x), where f(x) represent the y value. • F(x) is read as ‘y as a function of x’ • When you are given a function rule, you can evaluate the function at a given domain value to find the corresponding range value or vice versa.

18. ………. How to Evaluate a Function Rule To evaluate a function rule, substitute the value in for x and solve for y. Examples Evaluate the given function rules for f(2) which is a read as let x = 2. f(x)= x + 5 f(x)= 2x -1f(x)= -x + 2 y=-(2)+2 y= -2 + 2 y= 0 y=2(2)-1 y= 4 – 1 y= 3 y=(2)+ 5 y= 7

19. ………. When f(x) equals a number You can also be asked to let y equal a number and then solve for x. Examples Evaluate the given function rules for f(x) =2 which is a read as let y = 2. f(x)= x + 5 f(x)= 2x -1f(x)= -x + 7 2= -x+7 -5= -x x= 5 2 =2x -1 3 = 2x x= 3/2 2= x + 5 x = -3

20. Evaluating for multiple values • You can also be asked to find the range values for a given domain or vice versa. • This is the same as before, but now you’re evaluating the same function rule for more than one number. • The values that you are substituting in are x values, so they are apart of the domain. • The values you are generating are y-values, so they are apart of the range.

21. Example Find the range values of the function for the given domain. f(x) = -3x + 2 ; {-1, 0, 1, 2} y = -3x + 2 y = -3x + 2 y = -3x + 2 y = -3x + 2 y = -3(-1) + 2 y = -3(0) + 2 y = -3(1) + 2 y = -3(2) + 2 y = 3 + 2 y = 0 + 2 y = -3 + 2 y = -6 +2 y = 5 y = 2 y = -1 y = -4 The range values for the given domain are { 5, 2, -1, -4}. Steps Sub in each domain value in one @ a time. Solve for y in each List y values in braces.

22. Find the domain Find the domain values of the function for the given the range values. f(x) = 2x - 7 ; {-3, -2, 4} y = 2x -7 y = 2x -7 y = 2x - 7 -3 = 2x - 7 -2 = 2x -7 4 = 2x - 7 4 = 2x 5 = 2x 11= 2x x= 2 x = 5/2 x= 11/2 The domain values for the given range are { 2, 5/2, 11/2}.

23. Practice (you’ll need to hit the spacebar to pull up the next slide) 1. Find the range values of the function for the given domain. f(x) = 3x + 1 ; {-4, 0, 2} 2. Find the domain values of the function for the given range. f(x) = -2x + 3 ; {-5, -2, 6} Steps Sub in each value in one @ a time. Solve for the unknown in each List results in braces.

24. Answers 1. 2.