college algebra exam 3 material. ordered pair consists of: –t–two real numbers, –l–listed in...

College Algebra

Exam 3 Material

Ordered Pair

• Consists of:– Two real numbers,– Listed in a specific order,– Separated by a comma,– Enclosed within parentheses

• Examples of different ordered pairs:(7, 2)

(2, 7)

(3, -4)

first! fromdifferent ispair ordered Second :Note

Coordinates of an Ordered Pair

• The first number within an ordered pair is called the “x-coordinate”

• The second number within an ordered pair is called the “y-coordinate”

• Example, given the ordered pair (5, -2)– The x-coordinate is:

5– The y-coordinate is:

-2

Ordered Pairs & Interval Notation

• (2, 7) can have two meanings:– It can be an ordered pair.– It can be interval notation and have the meaning of all

real numbers between 2 and 7, but not including 2 and 7.

• Context will tell the difference, just like in the meaning of certain words.

• English Example, “bow”:– ribbon decoration on a package– an instrument for shooting an arrow

Rectangular Coordinate System

• Consists of:– Horizontal number line called “x-axis”– Vertical number line called “y-axis”– Intersecting at the zero points of each axis in a point ,

designated as the ordered pair (0, 0), and called the “origin”

– Forms a rectangular grid that can be used to show ordered pairs as points

axisx

axisy

Graphing Ordered Pair as a Point

• The x-coordinate gives horizontal directions from the origin• The y-coordinate gives vertical directions from the origin• When both directions have been followed, the point is located.• Example, graph: (-2, 3)

3,2

axisx

axisy

Rectangular Coordinate System

• Axes divide plane into four quadrants numbered counterclockwise from upper right

• The signs of the coordinates of a point are determined by its quadrant

• I :• II :• III :• IV :

I

III

III IV ,

, ,

,

Distance Formula

• The “distance” between two points (x1, y1) and (x2, y2) in a rectangular coordinate system is given by the formula:

Example: Distance between (2, -1) and (-3, 4)?

x1 = 2, y1 = -1, x2 = -3, y2 = 4

212

212 yyxx d

22 1-423- d

22 55 d

25502525 d

25 d

Midpoint Formula

• The “midpoint” of two points is a point exactly half way between the two points

• The formula for finding the coordinates of the midpoint of two points (x1, y1) and (x2, y2) is:

• Example: Midpoint of (2, -1) and (-3, 4)?

2

yy ,

2

xx mp 2121

2

41- ,

2

3-2 mp

2

3 ,

2

1 mp

Homework Problems

• Section: 2.1

• Page: 192

• Problems: All: 5 – 14

• MyMathLab Assignment 35 for practice

Ordered Pairs as Solutions to Equations in Two Variables

• An equation in two variables, “x” and “y,” requires a pair of numbers, one for “x” and one for “y”, to form a solution

• Example: Given the equation:2x + y = 5One solution consists of the pair:x = 2 and y = 1If it is agreed that “x” will be listed first and “y” second, this solution can be shown as the ordered pair:(2, 1)

Ordered Pairs as Solutions to Equations in Two Variables

Given: 2x + y = 5

Complete the missing coordinate to form ordered pair solutions:

( 1, __ )

( 3, __ )

( __, 5 )

( __, -3)

How many ordered pair solutions can be found?

31

04

number infiniteAn

Graphs ofEquations in Two Variables

• A “graph” of an equation in two variables is a “picture” in a rectangular coordinate system of all solutions to the equation

• A graph can be constructed by “point plotting,” finding enough (x, y) pairs to establish a pattern of points and then connecting the points with a smooth curve

• If the pattern indicates that the curve should extend beyond the graph grid, an arrow is placed on the curve where it exits the grid

Graph: 2x + y = 5

• Find (x, y) pairs by picking a number for “x” and solving for “y”, or vice versa

• Show solutions as (x, y) pairs• Plot each point• When pattern is seen

connect with smooth curve

with arrows at both ends• Solutions: 1,2 3,1 1,3

5,0 3,4 linestraight isPattern

Graph:

• Find and plot solutions (using a “T-chart” helps):

x

2 xy

y0 21 32 43 5

1 12 03 14 2

pattern shaped V

Homework Problems

• Section: 2.1

• Page: 194



Graphs vs. Form of Equation

• Experience will teach us that the general shape of a graph depends on the form of the equation in two variables

• Examples related to last section:– Equations of form: Ax + By = C will always

have a straight line graph– Equation of form: y = |mx + b| will always

have a V-shaped graph

Center Radius Equation of Circle

• Equation of the form:

• Will always have a graph that is a circle with center:

• and radius:

222 rkyhx

kh,

r

Center Radius Equation

• Considering:

• What would be the equation of a circle with center (-1, 2) and radius 3?

222 rkyhx

222 321 yx

921 22 yx

Center Radius Equation

• Given each of the following center radius equations, find the center and the radius: 1615 22 yx

137 22 yx

742 22 yx

:Radius

:Center

:Radius

:Center

:Radius

:Center

1,5

3,7

4,2

4

1

7

Homework Problems

• Section: 2.1

• Page: 194



General Equation of a Circle

• The equation will represent a real circle, if, when put in “center-radius form”,

Example:

• Don’t know yet, if it is a “real” circle, until written:

02 r

numbersrealareeanddcwhereedycxyx ,,,022

076222 yxyx

222 rkyhx

Converting General Equation to Center-Radius Equation

• “Complete the Square” twice, once on “x” and once on “y” (keep both sides balanced)Example:

• The “general equation” is a “real” circle since:

• Center and radius:

0172 r

076222 yxyx762 22 yyxx

9179612 22 yyxx

1731 22 yx

173,1 rand

Homework Problems

• Section: 2.1

• Page: 195



• MyMathLab Quiz 2.1 due for a grade on the date of our next class meeting

Relation

• Relation – any set of ordered pairs

Example: M = {(-3,2), (1,0), (4,-5)}

• Domain of a Relation – set of first members (“x coordinates”)

Example: Domain M = {-3, 1, 4}

• Range of a Relation – set of all second members (“y coordinates”)

Example: Range M = {2, 0, -5}

Equations in Two Variables

• Considered to be relations because solutions form a set of ordered pairs

Example:

• There are an infinite number of ordered pair solutions, but some are:

)2,4(),3,3(),2,2(),1,1(),0,0( and

xy

Domain ofEquations in Two Variables

• Set of all x’s for which y is a real number• May help to find domain by first solving for “y”

Example:

• Easy to see that x can be anything except -1:

Domain = ,11,

xyyx 3yxyx 3

yxx 13

yx

x

1

3

Range ofEquations in Two Variables

• Set of all y’s for which x is a real number• May help to find range by first solving for “x”

Example:

• Easy to see that y can be anything except 3:Range = ,33,

xyyx 3yxyx 3

yyx 3

yy

x

3

Function

• A special relation in which each x coordinate is paired with exactly one y coordinate

Example – only one of these is a function:

R = {(2,1), (3,-5), (2,3)}

Not a Function

S = {(3,2), (1,2), (-5,3)}

FUNCTION

Homework Problems

• Section: 2.2

• Page: 209



Dependent & Independent Variables in Functions

• Since every x is paired with exactly one y, we say that “y depends on x”

• For a function defined by an equation in two variables, x is the “independent variable” and y is the “dependent variable”

Functions Definedby Equations in Two Variables

• To determine if an equation in two variables is a function, solve for y and consider whether one x can give more than one y – if not, it is a function

Example – only one of these is a function:

• Solve each equation for y and analyze:

12

x

y

12

x

y

Example Continued

12

x

y

2xy

12

x

y

xy 2

xy

!FUNCTIONyonegivesxOne

!' FunctionaNotsytwogivesxOne

Function Notation

• Functions represented by equations in two variables are traditionally solved for y because doing so shows how y depends on x

Example – each of these is a different function:

12 xy

132 xxy

3 xy

Function Notation Continued

• When working with functions it is also traditional to replace y with the symbol f(x) or some variation using a letter other than f. This gives different functions different names:

Previous Example Written in “Function Notation”

12 xxf

132 xxxg

3 xxh

Function Notation Continued

• Function notation “f(x)” is read as “f of x” and means “the value of y for the given x”

Example:

If:

Then:

This means that for this f function, when x is -2, y is -7

13 xxf

7161232 f

Graphs ofRelations and Functions

• Can be accomplished by point plotting methods already discussed

• The graph of a relation will be a function if, and only if, every vertical line intersects the graph in at most one point (VERTICAL LINE TEST)

Example of Vertical Line TestWhich graph represents a function?

• Above: passes vertical line test – Function• Below: fails vertical line test – Not a Function

Practice Using Function Notation

• Given the functions f, g and h defined as follows, and remembering that f(2) means “the value of y in the f function when x is 2”, find the value of each function for the value of “x” shown:

If

If g = {(1,3),(-2,4),(2,5)} find g(-2)

g(-2) = 4

213 ffindxxxf 71281222 3 f

Practice Using Function Notation

Given the graph of y = h(x),

find h(-3)h(-3) = 1

Determining if Equation in Two Variables is a Function

• Solve the equation for y.

• If one x gives exactly one y, it is a function.

• If it is a function, it may be written in function notation by replacing y with f(x).

Example

Determine if the following equation defines a function, if so, write in function notation:

This is a function since one x gives one y

0462 2 xy

642 2 xy

32 2 xy

32 2 xxf

Increasing, Decreasing and Constant Functions

• A function is increasing over some interval of its domain if its graph goes up as x values go from left to right within the interval

• A function is decreasing over some interval of its domain if its graph goes down as x values go from left to right within the interval

• A function is constant over some interval of its domain if its graph is flat as x values go from left to right within the interval

Example

• Show the “interval of the domain” where the given function,

• is increasing: • is decreasing:• is constant:

]3,2[]2,(

),3[

Homework Problems

• Section: 2.2

• Page: 210

• Problems: Odd: 17 – 37, 41 – 63, 69 - 77


• MyMathLab Quiz 2.2 due for a grade on the date of our next class meeting

Linear Functions

• Any function that can be written in the form:

Example: f(x) = 3x - 1

• This is called the slope intercept form of a linear function (reason for name – explained later).

• The graph of a linear function will always be a non-vertical straight line

)( numbersrealarebandmwherebmxxf

Graphing Linear Functions

• Find any two ordered pairs that are solutions (chose two different x’s and find corresponding y’s)

• Plot the two points in a rectangular coordinate system.

• Draw a straight line connecting the two points with arrows on both ends.

Graphing a Linear Function

Graph f(x) = 3x – 1

Choose two values of “x” and calculate corresponding “y” values:

f(0) = 3(0) – 1 = -1 [ordered pair: (0,-1)]

f(2) = 3(2) – 1 = 5 [ordered pair: (2, 5)]

Linear FunctionsWith Horizontal Line Graphs

A linear function of the form:

can be simplified to:

This means y will always be the same (y = b) no matter the value of “x”

The graph will be a horizontal line through all the points that have a “y” value of “b”

)0( mwherebmxxf

bxf FunctionConstant

Example of Linear FunctionWith Horizontal Line Graph

Graph f(x) = -3

(Note: this function says no matter what value “x” is, the “y” value will be -3.)

3y

Homework Problems

• Section: 2.3

• Page: 221



Linear Relation

• A first degree polynomial equation in two variablesExamples:

• Standard form of a linear relation:

First example is in Standard Form (A=2, B=-3, C=6)Rest can be put in standard form, if desiredLast example is in standard form of a linear function

632 yxyx 48 3

2

1 xy

RelationLinear of Form Standard

FunctionLinear of Intercept) (Slope Form Standard

CByAx

Facts About Linear Relations

• Every linear relation,

is a linear function, can (by solving for y) be written in the form:

except when B = 0 (y term is missing)

CByAx

bmxxf

Example

• Write the linear relation in the standard form of a linear function:

432 yx

423 xy

3

4

3

2 xy

3

4

3

2 xxf

Facts About Linear Relations

• If a linear relation,

has B = 0 (the y term is missing), the relation is not a linear function, and it’s equation takes the form:

This equation has as solutions all the ordered pairs that have an x value of n (y can be any number) and its graph will be a vertical line.

CByAx

)(0 numberaxnxorCyAx

Example of Linear Relation with Vertical Line Graph

• Consider the linear relation:

• It can be written in the form:

• Two solutions are:

• Fails vertical line test -Not a linear function

4x

401 yx

2,45,4 and

4x

Homework Problems

• Section: 2.3

• Page: 221



Slope of a Line

• The slope, m, of a line is defined to be the ratio of the amount of vertical movement (rise) to the amount of horizontal movement (run) required to get from one point to another

rise

run

run

risemslope ,

negative? as described be line thisof slope theWhy would m-

run

rise

Slope of a Line Formula

• Given two points on a line, (x1, y1) and (x2, y2), the slope, m, can be calculated from the formula:

For a horizontal line y1 = y2 , so the slope of a horizontal line is always: ___

For a vertical line x1 = x2 , so the slope of a horizontal line is always: _________

run

rise

xx

yym

12

12

0

Undefined

Calculating Slope of a Line

• Find two points on the line and substitute into the formula

• Example: Find the slope of:

If x = 0, then y = ,

If y = 0, then x = ,

rise = 2, run = 3

3

2

03

20

m

632 yx2 2,0 3 0,3

:ispoint resulting:ispoint resulting

Homework Problems

• Section: 2.3

• Page: 222



Graphing a Line Given a Point and a Slope

• Given a point and a slope, m:– Plot the point– From the point, rise and run, according to the

value of m, to plot a second point– Connect the two points with a straight line with

arrows at both ends

1,1 yx

2,2 yx

1,1 yx

Example: Graph line through (-2, 1) with slope -2/3

Calculating Slope and y-intercept by “Solving for y”

• When a linear relation in two variables is solved for y it takes the form:where “m” is the slope, and “b” is the y-intercept

• In a previous example we used the slope formula to calculate slope: Calculating Slope of a Line A new approach is to find slope by solving for y:

What is “m”?

What is “b”?

bmxxforbmxy

632 yx623 xy

23

2 xy3

2

2

intercept"-y" therevealing ofbenefit added thegives and

formula theas slopefor valuesame gives y""for Solving

Graphing a Line inSlope-Intercept Form

• Write the equation in slope-intercept form

• Identify the y-intercept, b, and slope, m• Plot the y-intercept on the graph• From y-intercept, “rise” and “run” to

another point according to the ratio, m• Connect the points with a straight line

with arrows on each end

Example of Graphing a Line Written in Slope Intercept Form

• Graph:

y intercept is: 1, slope is:

Plot y intercept

From there rise -2, run 3 and plot

Connect points with a line

13

2 xy

3

2

Homework Problems

• Section: 2.3

• Page: 222



• MyMathLab Quiz 2.3 is due for a grade on the date of our next class meeting

Equations with Line GraphsRelationLinear of Form StandardCByAx

Function)(Constant Line Horizontal a isGraph - Case Specialny

FunctionLinear of FormIntercept Slope)0( mbmxy

Function) a(Not Line Vertical a isGraph - Case Specialnx

Point-Slope Equation of a Line

• Given a point (x1, y1) and a slope, m, the equation of the line through that point with that slope is:

y – y1 = m(x – x1)

Using Point Slope Equation

• Write the slope intercept equation of the line through (5, -1) with slope, m = -4/3

11 xxmyy

53

41 xy

3

20

3

41 xy

3

17

3

4 xy

Homework Problems

• Section: 2.4

• Page: 236



Parallel and Perpendicular Lines

• Two lines are “parallel” if they have the same slope, but different y-intercepts

• Two lines are “perpendicular” if they have slopes that are “negative reciprocals”

• Examples of slopes that are negative reciprocals:

3

2

2

3

55

1

11

Application Problem Involving Perpendicular Lines

• Write the slope intercept equation of a line that goes through the point (-1, 5) that is perpendicular to the line: x + 2y = 4

• First find slope of the line: x + 2y = 4 by solving for y: y = (-1/2)x + 2This line has slope, m = -1/2The perpendicular slope, m = 2Now use point-slope equation to write equation of line through (-1, 5) with slope 2:y – 5 = 2(x + 1)y = 2x + 7

Homework Problems

• Section: 2.4

• Page: 237




Basic Function Graphs

• The most basic functions include:• Identity function:• Squaring function:• Cubing function:• Square root function:• Cube root function:• Absolute value function:• Greatest integer function:

3)( xxf

2)( xxf xxf )(

xxf )(3)( xxf

xxf )(

xxf )(

Identify Function:

• Note: This is the slope intercept form of the linear function f(x) = mx + b, with m = 1 and b = 0

xfx00

11

xxf )(

Squaring Function:

• Note: This is a special case of the “quadratic function” that will be discussed later

xfx

2)( xxf

2101

2

0

4

4

1

1

Cubing Function:

xfx2101

2

0

1

1

3)( xxf

8

8

Square Root Function:

• Note: This is the first of the basic functions to have a restricted domain:

xfx

xxf )(

),0[

0 011

4 29 3

Cube Root Function:

• Does this function have a restricted domain like the last one?

xfx

3)( xxf

, NO,

8 21

01

1018 2

Absolute Value Function:

xfx4422

2

4

002

4

xxf )(

Greatest Integer Function:

• Note: This function pairs every real number x with the “greatest integer less than or equal to x”

• There is no restriction on the domain, but the range will be the set of all integers

xfx

xxf )(

x xf

2

7.2

2.3

9.3

4

1.4

5

2

3

4

4

4

5

5

9.0

3.0

0

1.0

7.0

1

3.1

0

0

0

1

1

1

2

Function" Step"

Include

Includet Don'

Homework Problems

• Section: 2.5

• Page: 249

• Problems: All: 7 – 14, 16, 33 – 35


Continuity of Functions

• A function is “continuous over some interval of its domain” if, over that interval, its graph can be drawn from one end to the other, using a pencil, without lifting the pencil from the paper

• The points where a function is not continuous are called “points of discontinuity”

Give domain and intervals of domain where these are

continuous: ,-

,- ) [0, ,-

,-

) [0, :Continuous

:Domain

:Continuous

:Domain

:Continuous

:Domain

This function is continuous over only short intervals of the domain. Give

domain and the points of discontinuity.

:itydiscontinu of Points

:Domain ) ,(-

integer.every at occur

Describe domain and the intervals of the domain where

the function is continuous:

:intervals in these Continuous

) (2,1] (-2,2- ,5- :Domain

) (2,1] (-2,2)- ,5[-

Homework Problems

• Section: 2.5

• Page: 249

• Problems: All: 1 – 6, 15


Piecewise Defined Functions

• These are functions that are defined by different rules for different intervals of their domain

• Example:

• To evaluate the function for specific values of “x”, use the formula for the interval that contains “x”

-2 xif

2 if 2

x

xxf(x)

5f 2f5 4

3f 3

2f 4

0f 0

GraphingPiecewise Defined Functions

• Plot enough points in each interval of the domain to establish the pattern in each interval

-2 xif

2 if 2

x

xxf(x)

:Graph

:Domain

:Range

:ityDiscontinu ofPoint

) ,(- ) 0,[)2,( 2x

:determined werepairs following theslide previous theFrom

4,2,0,0,4,2,3,3,5,5

Homework Problems

• Section: 2.5

• Page: 250

• Problems: Odd: 17 – 31



Transformations of Graphs

• Given a basic function graph, or any other, specific changes in the definition of the function lead to very similar graphs as follows:

• Given the graph of y = f(x), and assuming that h and v are positive, each of the following alterations to the function modifies the original graph as shown:

• y = - f(x)– Reflects original graph over x axis

• y = af(x)– Vertically stretches or squeezes original graph by a

factor of a


• y = f(x – h)– Horizontally translates the graph h units to the

right

• y = f(x + h)– Horizontally translates the graph h units to the

left


• y = f(x) – v– Vertically translates the graph v units

downward

• y = f(x) + v– Vertically translates the graph v units upward


• y = af(x – h) + v– Has the combination of effects previously

discussed – moves original graph horizontally right h units, shifts vertically upward v units and vertically stretches or squeezes by a factor of a

Example of Graph Transformations

xxf )(


• How does this transform the graph?

• Should reflect graph across the x axis:

xxf )(

blackin shown )( xxf



• Vertically stretches by a factor of 2:

xxf 2)(




• Vertically squeezes by a factor of ½ :

xxf2

1)(




• Moves it horizontally one unit to the right:

1)( xxf




• Translates it vertically down 4 units:

4)( xxf




• Horizontally translates 1 unit left

• Vertically translates 3 units down

• Vertically stretches by 2:

312)( xxf


Homework Problems

• Section: 2.6

• Page: 264

• Problems: All: 1 – 3, Odd: 5 – 11, 33 – 47


Even Functions

• A function f(x) is “even” if f(-x) = f(x) for every x in the domain

• Test to determine if a function is even:If substituting “–x” for “x” makes f(-x) = f(x), the function is “even”

• Even functions have graphs that are symmetric with respect to the y-axis– every line perpendicular to the y-axis that intersects

the graph at a distance of “d” from the y-axis will also intersect the graph at a distance of “d” on the other side of the y-axis

Determine if f(x) = x2 is even:2)( :Given xxf

)( xf 2x 2x )(xf function! EVEN

:axis-y respect to with Symmetric

Odd Functions

• A function f(x) is “odd” if f(-x) = - f(x) for every x in the domain

• Test to determine if a function is odd:If substituting “–x” for “x” makes f(-x) = - f(x), the function is “odd”

• Odd functions have graphs that are symmetric with respect to the origin– every line through the origin that intersects the graph

at a distance of “d” on one side of the origin will also intersect the graph at a distance of “d” on the other side of the origin

Determine if f(x) = x3 is odd:3)( :Given xxf

)( xf 3x 3x )(xf function! ODD

:origin respect to with Symmetric

Homework Problems

• Section: 2.6

• Page: 265




Resultant Functions from Operations on Functions

• Given functions f and g we can define sum, difference, product and quotient functions as follows:

• Sum function:

• Difference function:

• Product function:

• Quotient function:

xgxfxgf

xgxfxgf

xgxfxfg

0 ,

xg

xg

xfx

g

f

Domains of Sum, Difference, Product and Quotient Functions

• The domain of each of these functions is the intersection of their individual domains, with the exception that for the quotient function, those values of x are excluded for which g(x) = 0

• Note: Domains must be determined from individual functions not from looking only at the resultant function

Examples of Operations on Functions

• Given:

• Find definition and domain of:

xgf

xgf

xfg

x

g

f

xxgandx

xf

)( 1

1)(

: Domain

:Domain

g

f ,11, ,0

:Domain Domain gf ,11,0

xx

1

1

xx

1

1

1xx

Domain

Domain

Domain

Domain ,11,0 1

1

xx

,11,0

,11,0

,11,0

Examples of Operations on Functions

• Given information from previous slide:

xgf

xgf

xfg

x

g

f

xx

1

1

xx

1

1

1xx

1

1

xx

4gf

4fg

4

g

f

4gf3

7

3

5

3

2

6

1

Example: Determining Domain

• As previously noted: Domains must be determined from individual functions not from looking only at the resultant function

• Example:

.

x

xf1

2

x

xg1

2

:Domain f

:Domain g

,0

,0

xgf 4 :Domain g ,0

,- isdomain t think thayou wouldfunction resultant at theonly look you If

NOT

Homework Problems

• Section: 2.7

• Page: 276



Evaluating Resultant Functions from Graphs

xfy

xgy

1gf

2

g

f

2fg

0gf

2 11 gf

2

2

g

f3

22 gf 9

00 gf 0

Homework Problems

• Section: 2.7

• Page: 277



Difference Quotient

• If (x, f(x)) represents a point on the graph of y = f(x) and “h” is a positive number, then (x+h, f(x+h)) is a second point on the graph and, from the slope formula, the slope of the line joining these points is:

12

12

xx

yym

xhx

xfhxf )()(

h

xfhxf )()(

Calculus)in (ImportantQuotient Difference

Finding Difference Quotient

Given find difference quotient xxxf 26h

xfhxf )()(

hxhxhxf 26 hxhxhx 22 26

h

xxhxhxhx

h

xfhxf 222 66126)()(

h

xxhxhxhx 222 66126

h

hhxh 2612

h

hxh 16121612 hx

Homework Problems

• Section: 2.7

• Page: 278

• Problems: Odd: 33 – 39


Composition of Functions

• Given two functions f(x) and g(x), a third function, called the composition function:

may be defined as:

• The domain of the composition function is the set of all “x” in the domain of g , such that g(x) is in the domain of f

• Note: In general:

)(xgf

xgfxgf )(

)()( xfgxgf

Example of FormingComposite Functions

• Given:• Find:

32 2 xxgandxxf

)(xgf

xgfxgf )(

3 xf

232 x

962 2 xx

18122 2 xx )(xgf

Example to Show that Composition is Not Commutative• Given:• Find:

• Not the same!

32 2 xxgandxxf

)(xfg

xfgxfg )(

22xg

32 2 x )(xfg

Homework Problems

• Section: 2.7

• Page: 278

• Problems: Odd: 41 – 53, 57 – 63



college algebra exam 3 material. ordered pair consists of: –t–two real numbers, –l–listed in...

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