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# Relations and Functions

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### Relations and Functions

1. 1. Relations and Functions .
2. 2. What is a Relation? <ul><li>A relation is a set of ordered pairs. </li></ul><ul><li>When you group two or more points in a set, it is </li></ul><ul><li>referred to as a relation. When you want to show that a </li></ul><ul><li>set of points is a relation you list the points in braces. </li></ul><ul><li>For example, if I want to show that the points (-3,1) ; </li></ul><ul><li>(0, 2) ; (3, 3) ; & (6, 4) are a relation, it would be written </li></ul><ul><li>like this: </li></ul><ul><li>{(-3,1) ; (0, 2) ; (3, 3) ; (6, 4)} </li></ul><ul><li>……… . </li></ul>
3. 3. Domain and Range <ul><li>Each ordered pair has two parts, an x-value </li></ul><ul><li>and a y-value. </li></ul><ul><li>The x-values of a given relation are called the </li></ul><ul><li>Domain . </li></ul><ul><li>The y-values of the relation are called the </li></ul><ul><li>Range . </li></ul><ul><li>When you list the domain and range of </li></ul><ul><li>relation, you place each the domain and the </li></ul><ul><li>range in a separate set of braces. </li></ul><ul><li>……… . </li></ul>
4. 4. For Example, <ul><li>1. List the domain and the range of the relation </li></ul><ul><li>{(-3,1) ; (0, 2) ; (3, 3) ; (6, 4)} </li></ul><ul><li>Domain: { -3, 0, 3, 6} Range: {1, 2, 3, 4} </li></ul><ul><li>2. List the domain and the range of the relation </li></ul><ul><li>{(-3,3) ; (0, 2) ; (3, 3) ; (6, 4) ; ( 7, 7)} </li></ul><ul><li>Domain: {-3, 0, 3, 6, 7} Range: {3, 2, 4, 7} </li></ul><ul><li>Notice! Even though the number 3 is listed twice in the </li></ul><ul><li>relation, you only note the number once when you list the </li></ul><ul><li>domain or range! </li></ul><ul><li>……… . </li></ul>
5. 5. What is a Function? <ul><li>A function is a relation that assigns each </li></ul><ul><li>y-value only one x-value. </li></ul><ul><li>What does that mean? It means, in order for the </li></ul><ul><li>relation to be considered a function, there cannot be </li></ul><ul><li>any repeated values in the domain. </li></ul><ul><li>There are two ways to see if a relation is a function: </li></ul><ul><ul><ul><ul><ul><li>Vertical Line Test </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>Mappings </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>……… . </li></ul></ul></ul></ul></ul>
6. 6. Using the Vertical Line Test <ul><li>Use the vertical line test to check </li></ul><ul><li>if the relation is a function only if </li></ul><ul><li>the relation is already graphed. </li></ul><ul><li>Hold a straightedge (pen, ruler, </li></ul><ul><li>etc) vertical to your graph. </li></ul><ul><li>Drag the straightedge from left </li></ul><ul><li>to right on the graph. </li></ul><ul><li>3. If the straightedge intersects </li></ul><ul><li>the graph once in each spot , </li></ul><ul><li>then it is a function. </li></ul><ul><li>If the straightedge intersects the </li></ul><ul><ul><li>graph more than once in any </li></ul></ul><ul><ul><li>spot, it is not a function. </li></ul></ul><ul><li>A function! </li></ul>……… .
7. 7. Examples of the Vertical Line Test function function Not a function Not a function ……… .
8. 8. Mappings <ul><li>If the relation is not graphed, it is easier to use what is called a mapping . </li></ul><ul><li>When you are creating a mapping of a relation, you </li></ul><ul><li>draw two ovals. </li></ul><ul><li>In one oval, list all the domain values. </li></ul><ul><li>In the other oval, list all the range values. </li></ul><ul><li>Draw a line connecting the pairs of domain and range </li></ul><ul><li>values. </li></ul><ul><li>If any domain value ‘maps’ to two different range </li></ul><ul><li>values, the relation is not a function. </li></ul><ul><li>It’s easier than it sounds  </li></ul><ul><li>……… . </li></ul>
9. 9. Example of a Mapping <ul><li>Create a mapping of the following relation and state whether or not it is a function. </li></ul><ul><li>{(-3,1) ; (0, 2) ; (3, 3) ; (6, 4)} </li></ul><ul><li>Steps </li></ul><ul><li>Draw ovals </li></ul><ul><li>List domain </li></ul><ul><li>List range </li></ul><ul><li>Draw lines to connect </li></ul>-3 0 3 6 1 2 3 4 This relation is a function because each x-value maps to only one y-value. ……… .
10. 10. Another Mapping <ul><li>Create a mapping of the following relation and state whether or not it is a function. </li></ul><ul><li>{(-1,2) ; (1, 2) ; (5, 3) ; (6, 8)} </li></ul>-1 1 5 6 2 3 8 Notice that even though there are two 2’s in the range, you only list the 2 once. 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. ……… .
11. 11. Last Mapping <ul><li>Create a mapping of the following relation and state whether or not it is a function. </li></ul><ul><li>{(-4,-1) ; (-4, 0) ; (5, 1) ; (3, 9)} </li></ul>-4 5 3 -1 0 1 9 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. ……… . Make sure to list the (-4) only once!
12. 12. Vocabulary Review <ul><li>Relation : a set of order pairs. </li></ul><ul><li>Domain : the x-values in the relation. </li></ul><ul><li>Range : the y-values in the relation. </li></ul><ul><li>Function : a relation where each x-value is assigned (maps to) on one y-value. </li></ul><ul><li>Vertical Line Test : using a vertical straightedge to see if the relation is a function. </li></ul><ul><li>Mapping : a diagram used to see if the relation is a function. </li></ul><ul><li>……… . </li></ul>
13. 13. Practice (you will need to hit the spacebar to pull up the next slide) <ul><li>Complete the following questions and check your answers on the next slide. </li></ul><ul><li>Identify the domain and range of the following relations: </li></ul><ul><li>a. {(-4,-1) ; (-2, 2) ; (3, 1) ; (4, 2)} </li></ul><ul><li>b. {(0,-6) ; (1, 2) ; (7, -4) ; (1, 4)} </li></ul><ul><li>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. </li></ul><ul><li>a. {(-3,-3) ; (0, 6) ; (3, -3)} </li></ul><ul><li>b. {(0,6) ; (3, 3) ; (0, 0)} </li></ul><ul><li>Use a mapping to see if the following relations are functions: </li></ul><ul><ul><ul><ul><li> a. {(-4,-1) ; (-2, 2) ; (3, 1) ; (4, 2)} </li></ul></ul></ul></ul><ul><li>b. {(0,-6) ; (1, 2) ; (7, -4) ; (1, 4)} </li></ul>
14. 14. Answers (you will need to hit the spacebar to pull up the next slide) <ul><li>1a. Domain: {-4, -2, 3, 4} Range: {-2, 2, 1} </li></ul><ul><li>1b. Domain: {0, 1, 7} Range: {-6, 2, -4, 4} </li></ul><ul><li>2a. 2b. </li></ul><ul><li>3a. 3b. </li></ul>Function Not a Function Function Not a Function -4 -2 3 4 -1 2 1 0 1 7 -6 2 -4 4
15. 15. One more thing… <ul><li>The equation that represents a function is called a function rule . </li></ul><ul><li>A function rule is written with two variables, x </li></ul><ul><li>and y. </li></ul><ul><li>It can also be written in function notation , f(x), </li></ul><ul><li>but we’ll talk about that on Thursday. </li></ul><ul><li>When you are given a function rule, you can </li></ul><ul><li>evaluate the function at a given domain value to find </li></ul><ul><li>the corresponding range value. </li></ul><ul><li>……… . </li></ul>
16. 16. How to Evaluate a Function Rule <ul><li>To evaluate a function rule, substitute the </li></ul><ul><li>value in for x and solve for y. </li></ul><ul><li>Examples </li></ul><ul><li>Evaluate the given function rules for x = 2 </li></ul><ul><li> </li></ul><ul><li> </li></ul>y= x + 5 y= 2x -1 y= -x + 2 y=(2)+ 5 y= 7 y=2(2)-1 y= 4 – 1 y= 3 y=-(2)+2 y= -2 + 2 y= 0 ……… .
17. 17. Evaluating for a given domain <ul><li>You can also be asked to find the range </li></ul><ul><li>values for a given domain. </li></ul><ul><li>This is the same as before, but now </li></ul><ul><li>you’re evaluating the same function rule for </li></ul><ul><li>more than one number. </li></ul><ul><li>The values that you are substituting in are x </li></ul><ul><li>values, so they are apart of the domain. </li></ul><ul><li>The values you are generating are y-values, </li></ul><ul><li>so they are apart of the range. </li></ul><ul><li>……… . </li></ul>
18. 18. Example <ul><li>Find the range values of the function </li></ul><ul><li>for the given domain. </li></ul><ul><li>y = -3x + 2 ; {-1, 0, 1, 2} </li></ul><ul><li>y = -3x + 2 y = -3x + 2 y = -3x + 2 y = -3x + 2 </li></ul><ul><li>y = -3(-1) + 2 y = -3(0) + 2 y = -3(1) + 2 y = -3(2) + 2 </li></ul><ul><li>y = 3 + 2 y = 0 + 2 y = -3 + 2 y = -6 +2 </li></ul><ul><li>y = 5 y = 2 y = -1 y = -4 </li></ul><ul><li>The range values for the given domain are { 5, 2, -1, -4}. </li></ul><ul><li>……… . </li></ul><ul><li>Steps </li></ul><ul><li>Sub in each domain value in one @ a time. </li></ul><ul><li>Solve for y in each </li></ul><ul><li>List y values in braces. </li></ul>
19. 19. One more example <ul><li>Find the range values of the function </li></ul><ul><li>for the given domain. </li></ul><ul><li>y = 5x - 7 ; {-3, -2, 4} </li></ul><ul><li>y = 5x -7 y = 5x -7 y = 5x - 7 </li></ul><ul><li>y = 5(-3) - 7 y= 5(-2) -7 y = 5(4) - 7 </li></ul><ul><li>y = -15 - 7 y= -10 - 7 y= 20 - 7 </li></ul><ul><li>y= -22 y= -17 y= 13 </li></ul><ul><li>The range values for the given domain are </li></ul><ul><li>{ -22, -17, 13}. </li></ul>……… .
20. 20. Practice (you’ll need to hit the spacebar to pull up the next slide) <ul><li>1. Find the range values of the function </li></ul><ul><li>for the given domain. </li></ul><ul><li>y = 3x + 1 ; {-4, 0, 2} </li></ul><ul><li>2. Find the range values of the function </li></ul><ul><li>for the given domain. </li></ul><ul><li>y = -2x + 3 ; {-5, -2, 6} </li></ul><ul><li>Steps </li></ul><ul><li>Sub in each domain value in one @ a time. </li></ul><ul><li>Solve for y in each </li></ul><ul><li>List y values in braces. </li></ul>
21. 21. Answers 1. 2. y = 3x + 1 y = 3(-4) + 1 y = -12 + 1 y = -11 y = 3x + 1 y = 3(0) + 1 y = 0 + 1 y = 1 Ans. { -11, 1, 7} y = 3x + 1 y = 3(2) + 1 y = 6 + 1 y = 7 y = -2x + 3 y = -2(-5) + 3 y = 10 + 3 y = 13 y = -2x + 3 y = -2(-2) + 3 y = 4 +3 y = 7 Ans. { 13, 7, -9} y = -2x + 3 y = -2(6) + 3 y = -12 +3 y = -9