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Unit 1: Functions Lesson 1: Relations and Functions Learning Goals: I can determine if a relation is a function Unit 1: Functions Lesson 1: Relations and Functions So far, we have seen mathematical relationships written like this: These examples are relations: They are rules describing the • y = 3x + 1 relationship between the dependent • y = 2x2 -2 and independent variables. A relation is a connection (or 2 • y = x relationship) between two sets of numbers, such as height vs. time or • y = 5x cost vs. weight • Etc, etc. The Dependent Variable is: The Independent Variable is: Unit 1: Functions Lesson 1: Relations and Functions Example: The height, h, of an object thrown up in the air is dependent on the time, t. “h” is dependent on “t”, therefore h is the dependent variable and t is the independent variable. Unit 1: Functions Lesson 1: Relations and Functions These examples represent function notation and are read as, “ f of x”, or “f at x”. Unit 1: Functions Lesson 1: Relations and Functions Function notation represents a relation where there is only one unique value of the function (f) for any value of x. In other words, each x-value (independent variable) has only one yvalue (dependent variable) Unit 1: Functions Lesson 1: Relations and Functions How do you know whether something is a function? • If you put in a value for “x” and there is only one value for “y” it is a function. • If you put in a value for “x” and get more than one value for “y”, it is not a function. Unit 1: Functions Lesson 1: Relations and Functions Example: INPUT OUTPUT INPUT Camary Rav 4 Yaris Prius Toyota Toyota OUTPUT Camary Venza Rav 4 Sienna Yaris Corolla Prius Unit 1: Functions Lesson 1: Relations and Functions A function can be represented by: 1) A Table of Values 2) A Set of Ordered Pairs 3) A Mapping Diagram 4) A Graph 5) An Equation Unit 1: Functions Lesson 1: Relations and Functions Table of Values: It is a function if each x-value only corresponds to one y-value x -2 -1 0 -1 y 3 2 1 0 x 1 3 7 2 y 5 6 8 8 Unit 1: Functions Lesson 1: Relations and Functions Ordered Pairs: It is a function if for each xvalue there is only one y-value f = {(1,-4), (2, 5), (8, 9), (0, 6)} g = {(1, -3), (2, -3), (3, 0), (2, 0)} Unit 1: Functions Lesson 1: Relations and Functions Mapping Diagram: It is a function if the x-value points to only one y-value x 1 4 7 10 y x 11 1 8 0 10 -1 9 y 1 2 3 4 Unit 1: Functions Lesson 1: Relations and Functions Graph: It is a function if it passes the vertical line test. Vertical Line Test: Draw a vertical line through the graph. If the line crosses the graph more than once it is not a function. Unit 1: Functions Lesson 1: Relations and Functions Equation: Anything in the form y = mx + b is a function. Anything in the form y = ax2+ bx + c is a function. To check anything else, graph it! Unit 1: Functions Lesson 1: Relations and Functions Determine whether the following relations are functions or not a) y = 2x + 1 b) y = 2x2 - 3 c) x2 + y2 = 4 Unit 1: Functions Lesson 1: Relations and Functions Domain: The set of all the input values that are defined for a function. (Formerly referred to as the x-values or the independent variable.) Written from smallest to largest number. Range: The set of all the output values for the function. Can be determined by subbing in the values from the domain. (Formerly referred to as the y-values or the dependent variable.) Also written from smallest to largest number. Unit 1: Functions Lesson 1: Relations and Functions Unit 1: Functions Lesson 1: Relations and Functions Example: Write the domain and range for this function using set notation. x 1 3 7 2 y 5 6 8 8 Unit 1: Functions Lesson 1: Relations and Functions Example: Write the domain and range for this function using set notation. f = {(1,-4), (2,5), (8, 9), (0, 6)} Unit 1: Functions Lesson 1: Relations and Functions Example: Write the domain and range for this function using set notation. x 1 4 7 10 y 11 8 10 9 Unit 1: Functions Lesson 1: Relations and Functions Example: Write the domain and range for this function using set notation. Unit 1: Functions Lesson 1: Relations and Functions Practice Level 4: pg. 10-12 # 1 – 12, 14 Level 3: pg. 10-12 # 1 – 10, 14 Level 2: Pg. 10-12 #1-7, 14 Level 1: Pg. 10-12 #1 -3, 14