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