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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. The set of input values is called the domain. The set of output values is called the range.

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**Domain is the set of all x values.**

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

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**Find the Domain and Range of the following relation:**

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 x-axis y-axis origin quadrants Page 110

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**A Relation can be represented by a set of ordered pairs of the form (x,y)**

Quadrant II X<0, y>0 Quadrant I X>0, y>0 Origin (0,0) Quadrant IV X>0, y<0 Quadrant III X<0, y<0

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

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**Most equations have infinitely many solution points.**

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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**The collection of all solution points is the graph of the equation.**

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-1 y Page 112

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Ex 5. Graph y = x² - 5 x x² - 5 y -3 -2 -1 1 2 3

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**What are your questions?**

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Functions 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 output Page 116

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**Functions In order for a relationship to be a function…**

EVERY INPUT MUST HAVE AN OUTPUT TWO DIFFERENT INPUTS CAN HAVE THE SAME OUTPUT ONE INPUT CAN HAVE ONLY ONE OUTPUT INPUT Functions (DOMAIN) FUNCTION MACHINE OUTPUT (RANGE)

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**No two ordered pairs can have the same first coordinate**

Example 6 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)} No two ordered pairs can have the same first coordinate (and different second coordinates).

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**Identify the Domain and Range. Then tell if the relation is a function.**

Input Output 4 Function? Yes: each input is mapped onto exactly one output Domain = {-3, 1,3,4} Range = {3,1,-2}

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**Identify the Domain and Range. Then tell if the relation is a function.**

Input Output 4 Domain = {-3, 1,4} Range = {3,-2,1,4} Notice the set notation!!! Function? No: input 1 is mapped onto Both -2 & 1

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

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Is this a function? 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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**Use the vertical line test to visually check if the relation is a function.**

(4,4) (-3,3) (1,1) (1,-2) Function? No, Two points are on The same vertical line.

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**Use the vertical line test to visually check if the relation is a function.**

(-3,3) (1,1) (3,1) (4,-2) Function? Yes, no two points are on the same vertical line

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**Examples I’m going to show you a series of graphs.**

Determine whether or not these graphs are functions. You do not need to draw the graphs in your notes.

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YES! Function? #1

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**#2 Function? YES! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals**

Another cool function: abs(x) + 2sin(x)

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**#3 Function? NO! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals**

Another cool function: abs(x) + 2sin(x)

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**#4 Function? YES! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals**

Another cool function: abs(x) + 2sin(x)

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#5 Function? NO!

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YES! Function? #6 This is a piecewise function

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**Function? #7 NO! D: all reals R: [0, 1]**

Another cool function: y = sin(abs(x)) Y = sin(x) * abs(x)

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**#8 Function? NO! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals**

Another cool function: abs(x) + 2sin(x)

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YES! #9 Function?

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Function? #10 YES!

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Function? #11 NO! D: [-3, -1) U (-1, 3] R: {-1, 1}

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YES! Function? #12 D: [-3, -1) U (-1, 3] R: {-1, 1}

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

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**(x, y) (x, f(x)) (input, output) y = 6 – 3x f(x) = 6 – 3x x y x f(x)**

Before… Now… y = 6 – 3x f(x) = 6 – 3x x y x f(x) -2 -1 1 2 12 -2 -1 1 2 12 (x, y) (x, f(x)) 9 9 6 6 3 3 (input, output)

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**g(2) = 2 3 g(5) = Find g(2) and g(5).**

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

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**Consider the function h= { (-4, 0), (9,1), (-3, -2), (6,6), (0, -2)}**

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

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Example 9. f(x) = 2x2 – 3 Find f(0), f(-3), f(5a).

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**F(x) = 3x2 +1 Find f(0), f(-1), f(2a). f(0) = 1 f(-1) = 4**

Example 10. F(x) = 3x2 +1 Find f(0), f(-1), f(2a). f(0) = 1 f(-1) = 4 f(2a) = 12a2 + 1

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**The set of all real numbers that you can plug into the function.**

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

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**D: All real numbers except -3**

What is the domain? g(x) = -3x2 + 4x + 5 Ex. D: all real numbers x + 3 0 Ex. x -3 D: All real numbers except -3

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**What is the domain? h x ( ) = - 1 5 f x ( ) = + 1 2 x - 5 0 x + 2 0**

Ex. D: All real numbers except 5 Ex. f x ( ) = + 1 2 x + 2 0 D: All Real Numbers except -2

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**What are your questions?**

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2-1 Relations and Functions Analyze and Graph relations.

2-1 Relations and Functions Analyze and Graph relations.

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