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# 2.1 Relations and Functions

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### 2.1 Relations and Functions

1. 1. CHAPTER 2: FUNCTIONS, EQUATIONS, AND GRAPHS 2.1 Relations and Graphs
2. 2. RELATIONS <ul><li>A relation is a set of pairs of input and output values. </li></ul><ul><ul><li>There are four ways to represent relations </li></ul></ul>
3. 3. EXAMPLE: REPRESENTING RELATIONS <ul><li>The monthly average water temperature of the Gulf of Mexico in Key West, Florida varies during the year. In January, the average water temperature is 69 ° F, in February, 70 ° F, in March, 75 ° F and in April, 78 ° F. How can you represent this relation in four different ways? </li></ul>
4. 4. DEFINITIONS <ul><li>The domain of a relation is the set of inputs, also called the x-coordinates , of the ordered pairs. </li></ul><ul><li>The range is the set of outputs, also called the y-coordinates , of the ordered pairs. </li></ul>
5. 5. EXAMPLE: FIND THE DOMAIN AND RANGE OF EACH.
6. 6. FUNCTIONS <ul><li>A function is a relation in which each element of the domain corresponds to exactly one element of the range. </li></ul><ul><ul><li>When using a mapping diagram to represent a relation, a function has only one arrow from each element of the domain. </li></ul></ul><ul><ul><ul><li>Example: </li></ul></ul></ul>
7. 7. EXAMPLE: IS THE RELATION A FUNCTION?
8. 8. THE VERTICAL LINE TEST <ul><li>The vertical line test is used to test whether a graph represents a function. </li></ul><ul><li>The vertical line test states that if any vertical line passes through more than one point on the graph of a relation, then the relation is not a function. </li></ul>
9. 9. EXAMPLE: USE THE VERTICAL LINE TEST. WHICH GRAPH(S) REPRESENTS A FUNCTION?
10. 10. FUNCTIONS <ul><li>A function rule is an equation that represents an output value in terms of an input value. </li></ul><ul><ul><li>We write a function rule in function notation . </li></ul></ul>
11. 11. EXAMPLE: USING FUNCTION NOTATION <ul><li>For , what is the output for </li></ul><ul><li>the inputs: </li></ul>
12. 12. EXAMPLE: USING FUNCTION NOTATION <ul><li>For ,what is the output for </li></ul><ul><li>the inputs: </li></ul>
13. 13. EXAMPLE: <ul><li>Tickets to a concert are available online for \$35 each plus a handling fee of \$2.50. The total cost is a function of the number of tickets bought. What function rule models the cost of the concert tickets? Evaluate the function for 4 tickets. </li></ul>
14. 14. EXAMPLE <ul><li>You are buying bottles of a sports drink for a softball team. Each bottle costs \$1.19. What function rule models the cost of a purchase. Evaluate the function for 15 bottles. </li></ul>