Objective 1 Identify Domain and Range 2 Know

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

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

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 Ø Cartesian coordinate plane Ø x-axis Ø y-axis Ø origin Ø 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 III X<0, y<0 Quadrant IV X>0, y<0

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

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

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

Ex 4. Graph y = 3 x – 1. x 3 x-1 y Page 112

Ex 5. Graph x -3 -2 -1 0 1 2 3 y = x² - 5 y

What are your questions?

3. 3 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

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 (DOMAIN) FUNCTION MACHINE Functions OUTPUT (RANGE)

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

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

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. (-3, 3) (4, 4) (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 Ø I’m going to show you a series of graphs. Ø Determine whether or not these graphs are functions. Ø You do not need to draw the graphs in your notes.

#1 Function?

#2 Function?

#3 Function?

#4 Function?

#5 Function?

#6 Function?

#7 Function?

#8 Function?

#9 Function?

#10 Function?

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

Before… Now… y = 6 – 3 x f(x) = 6 – 3 x x y x f(x) -2 12 -1 9 0 6 1 3 2 0 (x, y) (input, output) (x, f(x))

Example 7 Find g(2) and g(5). g = {(1, 4), (2, 3), (3, 2), (4, -8), (5, 2)} g(2) = 3 g(5) = 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) = 2 2 x – 3 Find f(0), f(-3), f(5 a).

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

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

What is the domain? Ex. g(x) = -3 x 2 + 4 x + 5 D: all real numbers Ex. x+3 0 x -3 D: All real numbers except -3

What is the domain? Ex. 1 h( x ) = x -5 x-5 0 D: All real numbers except 5 Ex. f ( x) = 1 x+2 x + 2 0 D: All Real Numbers except -2

What are your questions?

Homework