Warm Up

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

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.

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}

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

3.2 Graphs

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

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

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

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

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.

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

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

What are your questions?

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

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)

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).

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

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

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

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)}

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

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.

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

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.

YES! Function? #1

#2 Function? YES! Y = 0.5x + 2 + 2sin(x) D: all reals R: all reals Another cool function: abs(x) + 2sin(x)

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

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

#5 Function? NO!

YES! Function? #6 This is a piecewise function

Function? #7 NO! D: all reals R: [0, 1] Another cool function: y = sin(abs(x)) Y = sin(x) * abs(x)

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

YES! #9 Function?

Function? #10 YES!

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

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

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

(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)

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) =

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).

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

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

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}

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

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

What are your questions?