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3-4 Functions 3-4 Functions Warm Up Problem of the Day Lesson Presentation Course Course 33 3-4 Functions Warm Up What three terms come next? 1. 9, 12, 15, 18, . . . 21, 24, 27 2. –8, –3, 2, 7, … 12, 17, 22 3. 9, 10, 12, 15, 19, … 24, 30, 37 Course 3 3-4 Functions Problem of the Day Sandra, Greg, and Michael team up for a competitive eating contest. If Sandra can eat 3 hot dogs per minute, Greg can eat 4 hot dogs per minute, and Michael can eat 5 1 hot dogs per minute, 2 how long will it take them to eat a combined total of 100 hot dogs? 8 minutes Course 3 3-4 Functions Learn to represent functions with tables, graphs, or equations. Course 3 3-4 Functions Vocabulary function input output domain range vertical line test Course 3 3-4 Functions A function is a rule that relates two quantities so that each input value corresponds to exactly one output value. The domain is the set of all possible input values, and the range is the set of all possible output values. Course 3 3-4 Functions Functions can be represented in many ways, including tables, graphs, and equations. If the domain of a function has infinitely many values, it is impossible to represent them all in a table, but a table can be used to show some of the values and to help in creating a graph. Course 3 3-4 Functions Additional Example 1: Finding Different Representations of a Function Make a table and a graph of y = 3 – x2. Make a table of inputs and outputs. Use the table to make a graph. x 3 – x2 y –2 3 – (–2)2 –1 –1 3 – (–1)2 2 0 3 – (0)2 3 1 3 – (1)2 2 2 3 – (2)2 –1 Course 3 3-4 Functions Check It Out: Example 1 Make a table and a graph of y = x + 1. Make a table of inputs and outputs. Use the table to make a graph. y x x+1 y 3 –1 –1 + 1 0 0 0+1 1 x –3 2 1 1+1 2 2 2+1 3 Course 3 3-4 Functions If a relationship is a function, each input has exactly one output. When the relationship is graphed, use the vertical line test. Place a vertical line on the graph. If the line intersects the graph at only one point, then the relationship is a function. If the line intersects the graph at more than one point, then the relationship is not a function. Course 3 3-4 Functions 11 Course 3 3-4 Functions Additional Example 2A: Identifying Functions Determine if the relationship represents a function. x y The input x = 2 has two outputs, y 2 3 = 3 and y = 6. The input x = 3 also has more than one output. 3 4 3 5 The relationship is not a function. 2 6 Course 3 3-4 Functions Additional Example 2B: Identifying Functions Determine if the relationship represents a function. The input x = 0 has two outputs, y = 2 and y = –2. Other x-values also have more than one y-value. The relationship is not a function. Course 3 3-4 Functions Additional Example 2C: Identifying Functions Determine if the relationship represents a function. y = x3 Make an input-output table and use it to graph y = x3. x y –2 (–2)3 = –8 –1 (–1)3 = –1 0 (0)3 = 0 1 (1)3 = 1 2 (2)3 = 8 Each input x has only one output y. The relationship is a function. Course 3 3-4 Functions Check It Out: Example 2A Determine if the relationship represents a function. x y Each input x has only one output 0 0 y. 1 1 The relationship is a function. 2 2 3 3 Course 3 3-4 Functions Check It Out: Example 2B Determine if the relationship represents a function. y Since the relationship is 2 linear there can only be one x output y for each input x. -2 2 -2 The relationship is a function. Course 3 3-4 Functions Check It Out: Example 2C Determine if the relationship represents a function. y=x–1 x x–1 y 0 0–1 –1 1 1–1 0 2 2–1 1 3 3–1 2 Each input x has only one output y. The relationship is a function. Course 3 3-4 Functions Lesson Quiz: Part I 1. Graph the function y = x2 – 3. y 4 2 x -4 -2 2 4 -2 -4 Course 3 3-4 Functions Lesson Quiz: Part II Determine if each relationship represents a function. 2. no 3. y = 3x + 5 yes Course 3