3.1 relations 3.2 graphs objective: find the cartesian product of two sets

3.1 Relations

3.2 Graphs

Objective: Find the Cartesian product of two sets.

Find the following Cartesian products.

A ,where { , }and {1,2}A X B A d e B

B ,where { , , }C X C C p q r

Objective: List ordered pairs from a Cartesian product that satisfy a given relation.

Any set of ordered pairs selected from a Cartesian product is a relation.

Objective: List the domain and the range of a relation.

C {(a, 1), (b, 2), (c, 3), (e, 2)}.

List the domain and the range of the relation

D {(2, 2), (1, 1), (1,2), (1, 3)}.

Objective: Use set-builder notation to define a relation.

Objective: Use set-builder notation to define a relation.

E Use the set {1, 2, 3, . . . , 10}.Find {x|5 < x < 7}.

F Use the set Q X Q, where Q = {2, 3, 4, 5}.Find {(x, y)|x > 2 and y > 3}.

Objective: Graph ordered pairs of a relation

Cartesian Coordinate System

Objective: Determine whether an ordered pair is a solution of an equation.

Solution: An ordered pair such that when the numbers are substituted for the variables, a true equation is produced

Determine whether the given ordered pairs are solutions to the equation y = 3x - 1:

G (7, 5)

H (7, 20)

I (0, 6)

Objective: Graph equations by plotting several solutions.

Graph the following relations

J 2 3y x 2K 3y x

HW #3.1-2Pg 108-109 1-29 Odd, 30-34Pg 114-115 29, 31, 37, 43-57

Pg 108-109 30b

Pg 108-109 30c

Pg 108-109 30d

Pg 114-115 37

Pg 108-109 30a

Pg 108-109 34

Pg 108-109 31c

Pg 114-115 37

HW Quiz #3.1-2Friday, April 21, 2023

Chapter 3Relations, Functions, and Graphs

3.3 Functions

Objective: Recognize functions and their graphs.

A relation where each member of the domain is paired with exactly one member of the range is a function.

Objective: Recognize functions and their graphs.

Which of the following relations are functions?

A

B

Objective: Recognize functions and their graphs.

Objective: Recognize functions and their graphs.

Function Not a Function

Which of the following relations are functions?

C D

Objective: Use function notation to find the value of functions.

FUNCTION MACHINE

( )f xPronounced “f of x”

Objective: Use function notation to find the value of functions.

FUNCTION MACHINE

Objective: Use function notation to find the value of functions.

Objective: Use function notation to find the value of functions.

Objective: Use function notation to find the value of functions.

2For the function defined by 3 27, evaluate:h h x x

E (5)h F ( 2)h

Consider {(0,2), ( 2,4), (1,0), ( 3,4)} find:g

G ( 2)g H (0)g

Objective: Find the domain of a function, given a formula for the function.

When the function in R X R is given by a formula, the domain is understood to be all real numbers that are acceptable replacements.

Finding the domain of a function 2 rules

1. Cannot let 0 be in the denominator

2. Cannot take a square root of a negative number

Objective: Find the domain of a function, given a formula for the function.

Find the domain of the following functions. State the domain using set-builder notation

4 2I 2f x x x 2J

( 2)( 4)

xp x

x x

HW #3.3-4Pg 120-121 1-27 odd, 30-36

Pg 125-126 3-9 Odd, 11, 17, 21, 25, 27, 36-42

HW Quiz #3.3-4Friday, April 21, 2023

Chapter 3Relations, Functions, and Graphs

3.4 Graphs of Linear Functions

3.5 Slope

Objective: Find the slope of a line containing a given pair of points.

Slope is the measure of how steep a line is

Objective: Find the slope of a line containing a given pair of points.

Slope is the measure of how steep a line is

Objective: Find the slope of a line containing a given pair of points.

Objective: Find the slope of a line containing a given pair of points.

Objective: Find the slope of a line containing a given pair of points.

Objective: Find the slope of a line containing a given pair of points.

Objective: Use the point-slope equation to find an equation of a line. .

HW #3.4-5Pg 125-126 3-9 Odd, 11, 17, 21,

25, 27, 36-42Pg 131-132 3-39 Every Third

Problem, 45-55

Chapter 3Relations, Functions, and Graphs

3.6 More Equations of Lines

Objective: Use the two point equation to find an equation of a line. .

Objective: Use the two point equation to find an equation of a line.

Objective: Use the two point equation to find an equation of a line.

Objective: Find the slope and y-intercept of a line, given the slope-intercept equation for the line.

Objective: Find the slope and y-intercept of a line, given the slope-intercept equation for the line.

Objective: Find the slope and y-intercept of a line, given the slope-intercept equation for the line.

Objective: Graph linear equations in slope-intercept form.

Objective: Graph linear equations in slope-intercept form.

Objective: Graph linear equations in slope-intercept form.

Chapter 3Relations, Functions, and Graphs

3.7 Parallel and Perpendicular lines

Objective: Determine if two lines are parallel or perpendicular or neither.

HW #3.6-7Pg 136-137 3-45 Every Third

Problem, 48-59

Pg 141-142 1-29 odd, 30-32

Pg 142 23 Pg 142 25 Pg 142 30a Pg 142 32

Pg 142 21 Pg 142 23 Pg 142 30b Pg 142 32

HW Quiz #3.7Friday, April 21, 2023

Chapter 3 Relations, Functions, and Graphs

3.9 More Functions

First class postage for letters or packages is a function of weight. For one ounce or less, the postage is $0.41. For each additional ounce or fraction of an ounce, $0.41 is due.

1. What is the postage for a 0.5 oz package?

2. What is the postage for a 0.7 oz package?

3. What is the postage for a 1 oz package?

4. What is the postage for a 1.5 oz package?

5. What is the postage for a 2 oz package?

6. What is the postage for a 2.5 oz package?

7. Sketch a graph of the weight of the package vs cost to ship

A step function has a graph which resembles a set of stair steps.

Objective: Graph special functions

Another example of a step function is the greatest integer function f(x) = [x].

The greatest integer function, f(x) = [x], is the greatest integer that is less than or equal to x.

Objective: Graph special functions

( ) [ ] 1f x x ( ) [ 1]f x x

Objective: Graph special functions

Finding the absolute value of a number can also be thought of in terms of a function, the absolute value function, f(x) = |x|.

Objective: Graph special functions

( ) | | 1f x x ( ) | 1|f x x

Sketch the graph of the following two functions

A ( ) [ 1]f x x B ( ) | 1|f x x

Objective: Find the composite of two functions

( )( ) ( ( ))f g x f g x

For ( ) 3and ( ) 3find :f x x g x x

C ( (2))f g D ( ( 4))g f

2For ( ) and ( ) 3find :p x x q x x

E ( ( ))p q x F ( ( ))q p x

For f(x) = 3x + b and g(x) = 2x – 7 find f(g(x))

For f(x) = px + d find f(f(x))

For f(x) = 2x + 6 and g(x) = 3x + b find b such that f(g(x)) = g(f(x))

Graph | | | | 1x y

HW #3.9Pg 150-151 1-25 Odd, 26-51

Pg 150 26a Pg 150 30 Pg 150 40 Pg 150 48

Pg 150 26b Pg 150 32 Pg 150 42 Pg 150 48

HW Quiz #3.9

HW Quiz #3.9Friday, April 21, 2023

Test ReviewObjective: List the domain and the range of a relation.

Objective: Recognize functions and their graphs.

Objective: Use function notation to find the value of functions.

Objective: Find the domain of a function, given a formula for the function.

Objective: Find the slope of a line containing a given pair of points.

Objective: Use the point-slope equation to find an equation of a line.

Objective: Graph linear equations in slope-intercept form.

Objective: Find the slope and y-intercept of a line, given the slope-intercept equation for the line.

Objective: Determine if two lines are parallel or perpendicular or neither.

Objective: Graph special functions

Objective: Find the composite of two functions

Objective: Find a linear function and use the equation to make predictions

Part 1

For f(x) = 3x + b and g(x) = 2x – 7 find f(g(x))

For f(x) = px + d find f(f(x))

For f(x) = 2x + 6 and g(x) = 3x + b find b such that f(g(x)) = g(f(x))

Given that f is a linear function with f(4)=-5 and f(0) = 3, write the equation that defines f.

Part 2

Show that the line containing the points (a, b) and (b, a) is perpendicular to the line y = x. Also show that the midpoint of (a, b) and (b, a) lies on the line y = x.

The equation 2x – y = C defines a family of lines, one line for each value of C. On one set of coordinate axes, graph the members of the family when C = -2, C= 0, and C= 4. Can you draw any conclusion from the graph about each member of the family? What about Cx +y = -4?

If two lines have the same slope but different x-intercepts, can they have the same y-intercept?

If two lines have the same y-intercept, but different slopes, can they have the same x-intercept?

The Greek method for finding the equation of a line tangent to a circle used the fact that at any point on a circle the line containing the center and the tangent line are perpendicular. Use this method to find the equation of the line tangent to the circle x2 + y2 = 9 at the point (1, 22).

Prove: If c d and a and b are not both zero, then ax + by =c and ax + by = d are parallel

HW #R-3Pg 157 1-30 Study all challenge

problems

Find the area of an equilateral triangle