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Warm Up

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Objective: 1. Identify Domain and Range 2. Know and use the Cartesian Plane 3. Graph equations using a chart 4. Determine if a Relation is a Function 5. Use the Vertical Line Test for Functions

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Relations A relation is a mapping, or pairing, of input values with output values. A relation is a mapping, or pairing, of input values with output values. The set of input values is called the domain. The set of input values is called the domain. The set of output values is called the range. The set of output values is called the range.

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Domain & Range Domain is the set of all x values. Range is the set of all y values. Example 1: Domain- D: {1, 2} Range- R: {1, 2, 3} {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)}

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Example 2: Find the Domain and Range of the following relation: {(a,1), (b,2), (c,3), (e,2)} Domain: {a, b, c, e}Range: {1, 2, 3} Page 107

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3.2 Graphs

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Cartesian Coordinate System Cartesian coordinate plane Cartesian coordinate plane x-axis x-axis y-axis y-axis origin origin quadrants quadrants Page 110

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A Relation can be represented by a set of ordered pairs of the form (x,y) Quadrant I X>0, y>0 Quadrant II X 0 Quadrant III X<0, y<0 Quadrant IV X>0, y<0 Origin (0,0)

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Plot: (-3,5)(-4,-2) (4,3)(3,-4)

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Every equation has solution points (points which satisfy the equation). 3x + y = 5 (0, 5), (1, 2), (2, -1), (3, -4) Some solution points: Most equations have infinitely many solution points. Page 111

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Ex 3. Determine whether the given ordered pairs are solutions of this equation. (-1, -4) and (7, 5); y = 3x -1 The collection of all solution points is the graph of the equation.

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Ex4. Graph y = 3x – 1. x 3x-1y Page 112

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Ex 5. Graph y = x² - 5 x x² - 5 y

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What are your questions?

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Functions Page 116 A relation as a function provided there is exactly one output for each input.A relation as a function provided there is exactly one output for each input. It is NOT a function if at least one input has more than one outputIt is NOT a function if at least one input has more than one output

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INPUT (DOMAIN) OUTPUT (RANGE) FUNCTION MACHINE In order for a relationship to be a function… EVERY INPUT MUST HAVE AN OUTPUT TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT Functions ONE INPUT CAN HAVE ONLY ONE OUTPUT

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Example 6 No two ordered pairs can have the same first coordinate (and different second coordinates). Which of the following relations are functions? R= {(9,10, (-5, -2), (2, -1), (3, -9)} S= {(6, a), (8, f), (6, b), (-2, p)} T= {(z, 7), (y, -5), (r, 7) (z, 0), (k, 0)}

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Identify the Domain and Range. Then tell if the relation is a function. Input Output Domain = {-3, 1,3,4} Range = {3,1,-2} Function? Yes: each input is mapped onto exactly one output

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Input Output Identify the Domain and Range. Then tell if the relation is a function. Domain = {-3, 1,4} Range = {3,-2,1,4} Function? No: input 1 is mapped onto Both -2 & 1 Notice the set notation!!!

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Look at example 1 on page 116 Do Try This a at the bottom of page 116 Do Try This a at the bottom of page 116

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1. {(2,5), (3,8), (4,6), (7, 20)} 2. {(1,4), (1,5), (2,3), (9, 28)} 3. {(1,0), (4,0), (9,0), (21, 0)}

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The Vertical Line Test If it is possible for a vertical line to intersect a graph at more than one point, then the graph is NOT the graph of a function. Page 117

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(-3,3) (4,4) (1,1) (1,-2) Use the vertical line test to visually check if the relation is a function. Function? No, Two points are on The same vertical line.

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(-3,3) (4,-2) (1,1) (3,1) Use the vertical line test to visually check if the relation is a function. Function? Yes, no two points are on the same vertical line

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Examples Im going to show you a series of graphs. Im going to show you a series of graphs. Determine whether or not these graphs are functions. Determine whether or not these graphs are functions. You do not need to draw the graphs in your notes. You do not need to draw the graphs in your notes.

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#1 Function?

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#2

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Function? #3

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Function? #4

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Function? #5

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#6 Function?

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#7

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Function? #8

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#9 Function?

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Function? #10

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Function? #11

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Function? #12

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f of x Input = x Output = f(x) = y Function Notation

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y = 6 – 3x x y f(x) = 6 – 3x x f(x) Before… Now… (x, y) (input, output) (x, f(x))

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Find g(2) and g(5). g = {(1, 4),(2,3),(3,2),(4,-8),(5,2)} g(2) = 3 g(5) = 2 Example 7

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Consider the function h= { (-4, 0), (9,1), (-3, -2), (6,6), (0, -2)} Example 8 Find h(9), h(6), and h(0).

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Example 9. f(x) = 2x 2 – 3 Find f(0), f(-3), f(5a).

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F(x) = 3x 2 +1 Example 10. Find f(0), f(-1), f(2a). f(0) = 1 f(-1) = 4 f(2a) = 12a 2 + 1

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The set of all real numbers that you can plug into the function. Domain D: {-3, -1, 0, 2, 4}

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g(x) = -3x 2 + 4x + 5 D: all real numbers Ex. What is the domain? x x -3 D: All real numbers except -3

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hx x () 1 5 x Ex. What is the domain? D: All real numbers except 5 D: All Real Numbers except -2 Ex. x fx x () 1 2

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What are your questions?

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