Functions and Their Graphs

The Cartesian Coordinate System and Straight linesEquations of LinesFunctions and Their GraphsThe Algebra of FunctionsLinear FunctionsQuadratic FunctionsFunctions and Mathematical ModelsFunctions and Their Graphs

2.1The Cartesian Coordinate System and Straight lines

We can represent real numbers geometrically by points on a real number, or coordinate, line:The Cartesian Coordinate System

The Cartesian Coordinate SystemThe Cartesian coordinate system extends this concept to a plane (two dimensional space) by adding a vertical axis.4321

1 2 3 4

The Cartesian Coordinate SystemThe horizontal line is called the x-axis, and the vertical line is called the y-axis.4321

1 2 3 4xy

The Cartesian Coordinate SystemThe point where these two lines intersect is called the origin.4321

1 2 3 4xyOrigin

The Cartesian Coordinate SystemIn the x-axis, positive numbers are to the right and negative numbers are to the left of the origin.4321

1 2 3 4xyPositive DirectionNegative Direction

The Cartesian Coordinate SystemIn the y-axis, positive numbers are above and negative numbers are below the origin.4321

1 2 3 4xyPositive DirectionNegative Direction

( 2, 4)(1, 2)(4, 3)The Cartesian Coordinate SystemA point in the plane can now be represented uniquely in this coordinate system by an ordered pair of numbers (x, y).4321

1 2 3 4xy(3, 1)

The Cartesian Coordinate SystemThe axes divide the plane into four quadrants as shown below.4321

1 2 3 4xyQuadrant I(+, +)Quadrant II(, +)Quadrant IV(+, )Quadrant III(, )

Slope of a Vertical LineLet L denote the unique straight line that passes through the two distinct points (x1, y1) and (x2, y2).If x1 = x2, then L is a vertical line, and the slope is undefined.(x1, y1)(x2, y2)yxL

Slope of a Nonvertical LineIf (x1, y1) and (x2, y2) are two distinct points on a nonvertical line L, then the slope m of L is given by(x1, y1)(x2, y2)yxLy2 y1 = yx2 x1 = x

Slope of a Nonvertical LineIf m > 0, the line slants upward from left to right.xLy = 1x = 1m = 1y

Slope of a Nonvertical LineIf m > 0, the line slants upward from left to right.yxLy = 2x = 1m = 2

m = 1Slope of a Nonvertical LineIf m < 0, the line slants downward from left to right.xLy = 1x = 1y

m = 2Slope of a Nonvertical LineIf m < 0, the line slants downward from left to right.yxLy = 2x = 1

123456(2, 5)ExamplesSketch the straight line that passes through the point (2, 5) and has slope 4/3.SolutionPlot the point (2, 5).A slope of 4/3 means that if x increases by 3, y decreases by 4.Plot the resulting point (5, 1).Draw a line through the two points.yxLy = 4x = 3654321

(5, 1)

ExamplesFind the slope m of the line that goes through the points (1, 1) and (5, 3). SolutionChoose (x1, y1) to be (1, 1) and (x2, y2) to be (5, 3). With x1 = 1, y1 = 1, x2 = 5, y2 = 3, we find

ExamplesFind the slope m of the line that goes through the points (2, 5) and (3, 5). SolutionChoose (x1, y1) to be (2, 5) and (x2, y2) to be (3, 5). With x1 = 2, y1 = 5, x2 = 3, y2 = 5, we find

2 1 1234ExamplesFind the slope m of the line that goes through the points (2, 5) and (3, 5). SolutionThe slope of a horizontal line is zero:yxL6

4321(2, 5)(3, 5)m = 0

Parallel LinesTwo distinct lines are parallel if and only if their slopes are equal or their slopes are undefined.

ExampleLet L1 be a line that passes through the points (2, 9) and (1, 3), and let L2 be the line that passes through the points ( 4, 10) and (3, 4). Determine whether L1 and L2 are parallel. SolutionThe slope m1 of L1 is given by

The slope m2 of L2 is given by

Since m1 = m2, the lines L1 and L2 are in fact parallel.

2.2Equations of Lines

Equations of LinesLet L be a straight line parallel to the y-axis.Then L crosses the x-axis at some point (a, 0) , with the x-coordinate given by x = a, where a is a real number.Any other point on L has the form (a, ), where is an appropriate number.The vertical line L can therefore be described asx = a(a, )yxL(a, 0)

Equations of LinesLet L be a nonvertical line with a slope m.Let (x1, y1) be a fixed point lying on L, and let (x, y) be a variable point on L distinct from (x1, y1).Using the slope formula by letting (x, y) = (x2, y2), we get

Multiplying both sides by x x1 we get

Point-Slope FormAn equation of the line that has slope m and passes through point (x1, y1) is given by

ExamplesFind an equation of the line that passes through the point (1, 3) and has slope 2.SolutionUse the point-slope form

Substituting for point (1, 3) and slope m = 2, we obtain

Simplifying we get

ExamplesFind an equation of the line that passes through the points (3, 2) and (4, 1).SolutionThe slope is given by

Substituting in the point-slope form for point (4, 1) and slope m = 3/7, we obtain

Perpendicular LinesIf L1 and L2 are two distinct nonvertical lines that have slopes m1 and m2, respectively, then L1 is perpendicular to L2 (written L1 L2) if and only if

ExampleFind the equation of the line L1 that passes through the point (3, 1) and is perpendicular to the line L2 described by

SolutionL2 is described in point-slope form, so its slope is m2 = 2.Since the lines are perpendicular, the slope of L1 must be m1 = 1/2Using the point-slope form of the equation for L1 we obtain

(a, 0)(0, b)Crossing the AxisA straight line L that is neither horizontal nor vertical cuts the x-axis and the y-axis at, say, points (a, 0) and (0, b), respectively.The numbers a and b are called the x-intercept and y-intercept, respectively, of L.yxLy-interceptx-intercept

Slope-Intercept FormAn equation of the line that has slope m and intersects the y-axis at the point (0, b) is given byy = mx + b

ExamplesFind the equation of the line that has slope 3 and y-intercept of 4.SolutionWe substitute m = 3 and b = 4 into y = mx + b and gety = 3x 4

ExamplesDetermine the slope and y-intercept of the line whose equation is 3x 4y = 8.SolutionRewrite the given equation in the slope-intercept form.

Comparing to y = mx + b, we find that m = and b = 2.So, the slope is and the y-intercept is 2.

Applied ExampleSuppose an art object purchased for $50,000 is expected to appreciate in value at a constant rate of $5000 per year for the next 5 years.Write an equation predicting the value of the art object for any given year.What will be its value 3 years after the purchase?Solution Let x = time (in years) since the object was purchasedy = value of object (in dollars)Then, y = 50,000 when x = 0, so the y-intercept is b = 50,000.Every year the value rises by 5000, so the slope is m = 5000.Thus, the equation must be y = 5000x + 50,000.After 3 years the value of the object will be $65,000:y = 5000(3) + 50,000 = 65,000

General Form of a Linear EquationThe equationAx + By + C = 0where A, B, and C are constants and A and B are not both zero, is called the general form of a linear equation in the variables x and y.

General Form of a Linear EquationAn equation of a straight line is a linear equation; conversely, every linear equation represents a straight line.

ExampleSketch the straight line represented by the equation3x 4y 12 = 0SolutionSince every straight line is uniquely determined by two distinct points, we need find only two such points through which the line passes in order to sketch it.For convenience, lets compute the x- and y-intercepts:Setting y = 0, we find x = 4; so the x-intercept is 4.Setting x = 0, we find y = 3; so the y-intercept is 3.Thus, the line goes through the points (4, 0) and (0, 3).

ExampleSketch the straight line represented by the equation3x 4y 12 = 0SolutionGraph the line going through the points (4, 0) and (0, 3).123456(0, 3)yxL1

1 2 3 4

(4, 0)

Equations of Straight LinesVertical line:x = aHorizontal line:y = bPoint-slope form: y y1 = m(x x1)Slope-intercept form: y = mx + bGeneral Form:Ax + By + C = 0

2.3Functions and Their Graphs

FunctionsA function f is a rule that assigns to each element in a set A one and only one element in a set B.The set A is called the domain of the function.It is customary to denote a function by a letter of the alphabet, such as the letter f.If x is an element in the domain of a function f, then the element in B that f associates with x is written f(x) (read f of x) and is called the value of f at x.The set B comprising all the values assumed by y = f(x) as x takes on all possible values in its domain is called the range of the function f.

ExampleLet the function f be defined by the rule Find: f(1)Solution:

ExampleLet the function f be defined by the rule Find: f( 2)Solution:

ExampleLet the function f be defined by the rule Find: f(a)Solution:

ExampleLet the function f be defined by the rule Find: f(a + h)Solution:

Applied ExampleThermoMaster manufactures an indoor-outdoor thermometer at its Mexican subsidiary.Management estimates that the profit (in dollars) realizable by ThermoMaster in the manufacture and sale of x thermometers per week is

Find ThermoMasters weekly profit if its level of production is:1000 thermometers per week. 2000 thermometers per week.

Applied ExampleSolutionWe have

The weekly profit by producing 1000 thermometers is

or $2,000.The weekly profit by producing 2000 thermometers is

or $7,000.

Determining the Domain of a FunctionSuppose we are given the function y = f(x).Then, the variable x is called the independent variable.The variable y, whose value depends on x, is called the dependent variable.To determine the domain of a function, we need to find what restrictions, if any, are to be placed on the independent variable x.In many practical problems, the domain of a function is dictated by the nature of the problem.

x16 2xApplied Example: PackagingAn open box is to be made from a rectangular piece of cardboard 16 inches wide by cutting away identical squares (x inches by x inches) from each corner and folding up the resulting flaps.

10 10 2x16xxx

Applied Example: PackagingAn open box is to be made from a rectangular piece of cardboard 16 inches wide by cutting away identical squares (x inches by x inches) from each corner and folding up the resulting flaps.

Find the expression that gives the volume V of the box as a function of x. What is the domain of the function?The dimensions of the resulting box are:x16 2x10 2x

x16 2xApplied Example: PackagingSolutiona.The volume of the box is given by multiplying its dimensions (length width height), so:10 2x

Applied Example: PackagingSolutionb.Since the length of each side of the box must be greater than or equal to zero, we see that

must be satisfied simultaneously. Simplified:

All three are satisfied simultaneously provided that:

Thus, the domain of the function f is the interval [0, 5].

More ExamplesFind the domain of the function:

SolutionSince the square root of a negative number is undefined, it is necessary that x 1 0.Thus the domain of the function is [1,).


SolutionOur only constraint is that you cannot divide by zero, so

Which means that

Or more specifically x 2 and x 2.Thus the domain of f consists of the intervals ( , 2), (2, 2), (2, ).


SolutionHere, any real number satisfies the equation, so the domain of f is the set of all real numbers.

Graphs of FunctionsIf f is a function with domain A, then corresponding to each real number x in A there is precisely one real number f(x).

Thus, a function f with domain A can also be defined as the set of all ordered pairs (x, f(x)) where x belongs to A.

The graph of a function f is the set of all points (x, y) in the xy-plane such that x is in the domain of f and y = f(x).

ExampleThe graph of a function f is shown below:xyDomainRangexy(x, y)

ExampleThe graph of a function f is shown below:What is the value of f(2)?xy4321

12

(2, 2) 1234 5678

ExampleThe graph of a function f is shown below:What is the value of f(5)?xy(5, 3)4321

12

1234 5678

ExampleThe graph of a function f is shown below:What is the domain of f(x)?xy4321

12

Domain: [1,8] 1234 5678

ExampleThe graph of a function f is shown below:What is the range of f(x)?xy4321

12

Range: [2,4] 1234 5678

Example: Sketching a GraphSketch the graph of the function defined by the equation y = x2 + 1SolutionThe domain of the function is the set of all real numbers.Assign several values to the variable x and compute the corresponding values for y:

xy3102512011225310

3 2 11 23108642

Example: Sketching a GraphSketch the graph of the function defined by the equation y = x2 + 1SolutionThe domain of the function is the set of all real numbers.Then plot these values in a graph: xy

xy3102512011225310

3 2 11 23108642

Example: Sketching a GraphSketch the graph of the function defined by the equation y = x2 + 1SolutionThe domain of the function is the set of all real numbers.And finally, connect the dots: xy

xy3102512011225310

Example: Sketching a GraphSketch the graph of the function defined by the equation

SolutionThe function f is defined in a piecewise fashion on the set of all real numbers.In the subdomain ( , 0), the rule for f is given by In the subdomain [0, ), the rule for f is given by


SolutionSubstituting negative values for x into , whilesubstituting zero and positive values into we get:

xy332211001121.4131.73


SolutionPlotting these data and graphing we get: 3 2 11 23321xy

xy332211001121.4131.73

The Vertical Line TestA curve in the xy-plane is the graph of a function y = f(x) if and only if each vertical line intersects it in at most one point.

ExamplesDetermine if the curve in the graph is a function of x:

SolutionThe curve is indeed a function of x, because there is one and only one value of y for any given value of x.xy


SolutionThe curve is not a function of x, because there is more than one value of y for some values of x.xy


SolutionThe curve is indeed a function of x, because there is one and only one value of y for any given value of x.xy

2.4The Algebra of Functions

The Sum, Difference, Product and Quotient of FunctionsConsider the graph below:R(t) denotes the federal government revenue at any time t.S(t) denotes the federal government spending at any time t.199019921994199619982000200018001600140012001000tyBillions of DollarsYeary = S(t)y = R(t)tR(t)S(t)

D(t) = R(t) S(t)The Sum, Difference, Product and Quotient of FunctionsConsider the graph below:The difference R(t) S(t) gives the budget deficit (if negative) or surplus (if positive) in billions of dollars at any time t.199019921994199619982000200018001600140012001000tyBillions of DollarsYeary = S(t)y = R(t)tR(t)S(t)

The Sum, Difference, Product and Quotient of FunctionsThe budget balance D(t) is shown below:D(t) is also a function that denotes the federal government deficit (surplus) at any time t. This function is the difference of the two functions R and S.D(t) has the same domain as R(t) and S(t).199219941996199820004002000 200 400 tyBillions of DollarsYeart D(t)y = D(t)

The Sum, Difference, Product and Quotient of FunctionsMost functions are built up from other, generally simpler functions.For example, we may view the function f(x) = 2x + 4 as the sum of the two functions g(x) = 2x and h(x) = 4.

The Sum, Difference, Product and Quotient of FunctionsLet f and g be functions with domains A and B, respectively.The sum f + g, the difference f g, and the product fg of f and g are functions with domain A B and rule given by(f + g)(x) = f(x) + g(x)Sum(f g)(x) = f(x) g(x)Difference (fg)(x) = f(x)g(x)Product

The quotient f/g of f and g has domain A B excluding all numbers x such that g(x) = 0 and rule given by Quotient

ExampleLet and g(x) = 2x + 1. Find the sum s, the difference d, the product p, and the quotient q of the functions f and g.SolutionSince the domain of f is A = [1,) and the domain of g is B = ( , ), we see that the domain of s, d, and p is A B = [1,).The rules are as follows:

ExampleLet and g(x) = 2x + 1. Find the sum s, the difference d, the product p, and the quotient q of the functions f and g.SolutionThe domain of the quotient function is [1,) together with the restriction x .Thus, the domain is [1, ) U ( ,). The rule is as follows:

Applied Example: Cost FunctionsSuppose Puritron, a manufacturer of water filters, has a monthly fixed cost of $10,000 and a variable cost of 0.0001x2 + 10x(0 x 40,000)dollars, where x denotes the number of filters manufactured per month.Find a function C that gives the total monthly cost incurred by Puritron in the manufacture of x filters.

Applied Example: Cost FunctionsSolutionPuritrons monthly fixed cost is always $10,000, so it can be described by the constant function: F(x)= 10,000The variable cost can be described by the function:V(x)= 0.0001x2 + 10xThe total cost is the sum of the fixed cost F and the variable cost V: C(x)= V(x) + F(x) = 0.0001x2 + 10x + 10,000 (0 x 40,000)

Applied Example: Cost FunctionsLets now consider profitsSuppose that the total revenue R realized by Puritron from the sale of x water filters is given by R(x)= 0.0005x2 + 20x (0 x 40,000)FindThe total profit function for Puritron.The total profit when Puritron produces 10,000 filters per month.

Applied Example: Cost FunctionsSolutionThe total profit P realized by the firm is the difference between the total revenue R and the total cost C: P(x)= R(x) C(x) = ( 0.0005x2 + 20x) ( 0.0001x2 + 10x + 10,000)= 0.0004x2 + 10x 10,000The total profit realized by Puritron when producing 10,000 filters per month isP(x)= 0.0004(10,000)2 + 10(10,000) 10,000 = 50,000or $50,000 per month.

The Composition of Two FunctionsAnother way to build a function from other functions is through a process known as the composition of functions.Consider the functions f and g:

Evaluating the function g at the point f(x), we find that:

This is an entirely new function, which we could call h:

The Composition of Two FunctionsLet f and g be functions.Then the composition of g and f is the function ggf (read g circle f ) defined by (ggf )(x) = g(f(x))

The domain of ggf is the set of all x in the domain of f such that f(x) lies in the domain of g.

ExampleLetFind:The rule for the composite function ggf.The rule for the composite function fgg.SolutionTo find ggf, evaluate the function g at f(x):

To find fgg, evaluate the function f at g(x):

Applied Example: Automobile PollutionAn environmental impact study conducted for the city of Oxnard indicates that, under existing environmental protection laws, the level of carbon monoxide (CO) present in the air due to pollution from automobile exhaust will be 0.01x2/3 parts per million when the number of motor vehicles is x thousand.A separate study conducted by a state government agency estimates that t years from now the number of motor vehicles in Oxnard will be 0.2t2 + 4t + 64 thousand.Find:An expression for the concentration of CO in the air due to automobile exhaust t years from now.The level of concentration 5 years from now.

Applied Example: Automobile PollutionSolutionPart (a):The level of CO is described by the function g(x) = 0.01x2/3 where x is the number (in thousands) of motor vehicles.In turn, the number (in thousands) of motor vehicles is described by the functionf(t) = 0.2t2 + 4t + 64 where t is the number of years from now.Therefore, the concentration of CO due to automobile exhaust t years from now is given by(ggf )(t) = g(f(t)) = 0.01(0.2t2 + 4t + 64)2/3

Applied Example: Automobile PollutionSolutionPart (b):The level of CO five years from now is:(ggf )(5)= g(f(5)) = 0.01[0.2(5)2 + 4(5) + 64]2/3= (0.01)892/3 0.20 or approximately 0.20 parts per million.

2.5Linear Functions

Linear FunctionThe function f defined by

where m and b are constants, is called a linear function.

Applied Example: Linear DepreciationA Web server has an original value of $10,000 and is to be depreciated linearly over 5 years with a $3000 scrap value.Find an expression giving the book value at the end of year t.What will be the book value of the server at the end of the second year?What is the rate of depreciation of the server?

Applied Example: Linear DepreciationSolutionLet V(t) denote the Web servers book value at the end of the tth year. V is a linear function of t.To find an equation of the straight line that represents the depreciation, observe that V = 10,000 when t = 0; this tells us that the line passes through the point (0, 10,000).Similarly, the condition that V = 3000 when t = 5 says that the line also passes through the point (5, 3000).Thus, the slope of the line is given by

Applied Example: Linear DepreciationSolutionUsing the point-slope form of the equation of a line with point (0, 10,000) and slope m = 1400, we obtain the required expression

The book value at the end of the second year is given by

or $7200.The rate of depreciation of the server is given by the negative slope of the depreciation line m = 1400, so the rate of depreciation is $1400 per year.

Applied Example: Linear DepreciationSolutionThe graph of V is:123456(0, 10,000)Vt(5, 3000)10,0003000

Cost, Revenue, and Profit FunctionsLet x denote the number of units of a product manufactured or sold.Then, the total cost function isC(x)=Total cost of manufacturing x units of the productThe revenue function isR(x)=Total revenue realized from the sale of x units of the productThe profit function isP(x)=Total profit realized from manufacturing and selling x units of the product

Applied Example: Profit FunctionPuritron, a manufacturer of water filters, has a monthly fixed cost of $20,000, a production cost of $20 per unit, and a selling price of $30 per unit.Find the cost function, the revenue function, and the profit function for Puritron.SolutionLet x denote the number of units produced and sold.Then,

Finding the Point of IntersectionSuppose we are given two straight lines L1 and L2 with equations y = m1x + b1 and y = m2x + b2 (where m1, b1, m2, and b2 are constants) that intersect at the point P(x0, y0).The point P(x0, y0) lies on the line L1 and so satisfies the equation y = m1x + b1.The point P(x0, y0) also lies on the line L2 and so satisfies y = m2x + b2 as well.Therefore, to find the point of intersection P(x0, y0) of the lines L1 and L2, we solve for x and y the system composed of the two equationsy = m1x + b1 and y = m2x + b2

ExampleFind the point of intersection of the straight lines that have equationsy = x + 1 and y = 2x + 4SolutionSubstituting the value y as given in the first equation into the second equation, we obtain

Substituting this value of x into either one of the given equations yields y = 2.Therefore, the required point of intersection is (1, 2).

11 2 3 4554321ExampleFind the point of intersection of the straight lines that have equationsy = x + 1 and y = 2x + 4SolutionThe graph shows the point of intersection (1, 2) of the two lines:yx(1, 2)L1L2

Applied Example: Break-Even LevelPrescott manufactures its products at a cost of $4 per unit and sells them for $10 per unit.If the firms fixed cost is $12,000 per month, determine the firms break-even point.SolutionThe revenue function R and the cost function C are given respectively by

Setting R(x) = C(x), we obtain

Applied Example: Break-Even LevelPrescott manufactures its products at a cost of $4 per unit and sells them for $10 per unit.If the firms fixed cost is $12,000 per month, determine the firms break-even point.SolutionSubstituting x = 2000 into R(x) = 10x gives

So, Prescotts break-even point is 2000 units of the product, resulting in a break-even revenue of $20,000 per month.

2.6Quadratic Functions

Quadratic FunctionsA quadratic function is one of the form

where a, b, and c are constants and a 0.For example, the function

is quadratic, with a = 2, b = 4, and c = 3.

Quadratic FunctionsBelow is the graph of the quadratic function

The graph of a quadratic function is a curve called a parabola that opens upward or downward. 2 1 12 34108642

xyParabola

Quadratic FunctionsThe parabola is symmetric with respect to a vertical line called the axis of symmetry.The axis of symmetry also passes through the lowest or highest point of the parabola, which is called the vertex of the parabola. 2 1 12 34108642

xyAxis of symmetryVertex (1, 1)Parabola

Quadratic FunctionsWe can use these properties to help us sketch the graph of a quadratic function.Suppose we want to sketch the graph of

If we complete the square in x, we obtain

Note that (x 1)2 is nonnegative: it equals to zero when x = 1 and is greater than zero if x 1. Thus, we see that f(x) 2 for all values of x. This tells us the vertex of the parabola is the point (1, 2).

Quadratic FunctionsWe know the vertex of the parabola is the point (1, 2) and that it is the minimum point of the graph, since f(x) 2 for all values of x.Thus, the graph of f(x) = 3x2 6x +1 looks as follows: 2 2 442 2xyVertex (1, 2)

Properties of Quadratic FunctionsGiven f(x) = ax2 + bx +c (a 0)

The domain of f is the set of all real numbers.If a > 0, the parabola opens upward, and if a < 0, it opens downward.The vertex of the parabola is The axis of symmetry of the parabola is The x-intercepts (if any) are found by solving f(x) = 0. The y-intercept is f(0) = c.

ExampleGiven the quadratic function f(x) = 2x2 + 5x 2 Find the vertex of the parabola.Find the x-intercepts (if any) of the parabola.Sketch the parabola.SolutionHere a = 2, b = 5, and c = 2. therefore, the x-coordinate of the vertex of the parabola is

The y-coordinate of the vertex is therefore given by

Thus, the vertex of the parabola is the point

ExampleGiven the quadratic function f(x) = 2x2 + 5x 2 Find the vertex of the parabola.Find the x-intercepts (if any) of the parabola.Sketch the parabola.SolutionFor the x-intercepts of the parabola, we solve the equation

using the quadratic formula with a = 2, b = 5, and c = 2. We find

Thus, the x-intercepts of the parabola are 1/2 and 2.

ExampleGiven the quadratic function f(x) = 2x2 + 5x 2 Find the vertex of the parabola.Find the x-intercepts (if any) of the parabola.Sketch the parabola. SolutionThe sketch: 1 1 21

1 2xyVertexx-interceptsy-intercept

Some Economic ModelsPeoples decision on how much to demand or purchase of a given product depends on the price of the product:

The higher the price the less they want to buy of it.

A demand function p = d(x) can be used to describe this.

Some Economic ModelsSimilarly, firms decision on how much to supply or produce of a product depends on the price of the product:

The higher the price, the more they want to produce of it.

A supply function p = s(x) can be used to describe this.

Some Economic ModelsThe interaction between demand and supply will ensure the market settles to a market equilibrium:

This is the situation at which quantity demanded equals quantity supplied.

Graphically, this situation occurs when the demand curve and the supply curve intersect: where d(x) = s(x).

Applied Example: Supply and DemandThe demand function for a certain brand of bluetooth wireless headset is given by

The corresponding supply function is given by

where p is the expressed in dollars and x is measured in units of a thousand. Find the equilibrium quantity and price.

Applied Example: Supply and DemandSolutionWe solve the following system of equations:

Substituting the second equation into the first yields:

Thus, either x = 400/9 (but this is not possible), or x = 20. So, the equilibrium quantity must be 20,000 headsets.

Applied Example: Supply and DemandSolutionThe equilibrium price is given by:

or $40 per headset.

2.7Functions and Mathematical Models

Mathematical ModelsAs we have seen, mathematics can be used to solve real-world problems.We will now discuss a few more examples of real-world phenomena, such as:The solvency of the U.S. Social Security trust fundGlobal warming

Mathematical ModelingRegardless of the field from which the real-world problem is drawn, the problem is analyzed using a process called mathematical modeling.The four steps in this process are:Real-world problemMathematical modelSolution of real-world problemSolution of mathematical modelFormulateInterpretSolveTest

Modeling With Polynomial FunctionsA polynomial function of degree n is a function of the form

where n is a nonnegative integer and the numbers a0, a1, . an are constants called the coefficients of the polynomial function.Examples:The function below is polynomial function of degree 5:

Modeling With Polynomial FunctionsA polynomial function of degree n is a function of the form

where n is a nonnegative integer and the numbers a0, a1, . an are constants called the coefficients of the polynomial function.Examples:The function below is polynomial function of degree 3:

Applied Example: Global WarmingThe increase in carbon dioxide (CO2) in the atmosphere is a major cause of global warming.Below is a table showing the average amount of CO2, measured in parts per million volume (ppmv) for various years from 1958 through 2007:

Year195819701974197819851991199820032007Amount315325330335345355365375380

Applied Example: Global WarmingBelow is a scatter plot associated with these data:1020304050380360340320t (years)y (ppmv)

Year195819701974197819851991199820032007Amount315325330335345355365375380

Applied Example: Global WarmingA mathematical model giving the approximate amount of CO2 is given by:1020304050380360340320t (years)y (ppmv)

Year195819701974197819851991199820032007Amount315325330335345355365375380

Applied Example: Global WarmingUse the model to estimate the average amount of atmospheric CO2 in 1980 (t = 23).Assume that the trend continued and use the model to predict the average amount of atmospheric CO2 in 2010.

Year195819701974197819851991199820032007Amount315325330335345355365375380

Applied Example: Global WarmingSolutionThe average amount of atmospheric CO2 in 1980 is given by

or approximately 338 ppmv.Assuming that the trend will continue, the average amount of atmospheric CO2 in 2010 will be

Year195819701974197819851991199820032007Amount315325330335345355365375380

Applied Example: Social Security Trust FundThe projected assets of the Social Security trust fund (in trillions of dollars) from 2008 through 2040 are given by:

The scatter plot associated with these data is:51015202530642t (years)y ($trillion)

Year200820112014201720202023202620292032203520382040 Assets2.43.24.04.75.35.75.95.64.93.61.70

The projected assets of the Social Security trust fund (in trillions of dollars) from 2008 through 2040 are given by:

A mathematical model giving the approximate value of assets in the trust fund (in trillions of dollars) is:51015202530642t (years)y ($trillion)Applied Example: Social Security Trust Fund

Year200820112014201720202023202620292032203520382040 Assets2.43.24.04.75.35.75.95.64.93.61.70

The first baby boomers will turn 65 in 2011. What will be the assets of the Social Security trust fund at that time?The last of the baby boomers will turn 65 in 2029. What will the assets of the trust fund be at the time?Use the graph of function A(t) to estimate the year in which the current Social Security system will go broke.Applied Example: Social Security Trust Fund

Year200820112014201720202023202620292032203520382040 Assets2.43.24.04.75.35.75.95.64.93.61.70

SolutionThe assets of the Social Security fund in 2011 (t = 3) will be:

or approximately $3.18 trillion.The assets of the Social Security fund in 2029 (t = 21) will be:

or approximately $5.59 trillion.Applied Example: Social Security Trust Fund

Year200820112014201720202023202620292032203520382040 Assets2.43.24.04.75.35.75.95.64.93.61.70

SolutionThe graph shows that function A crosses the t-axis at about t = 32, suggesting the system will go broke by 2040:51015202530642y ($trillion)t (years)Applied Example: Social Security Trust Fund

Year200820112014201720202023202620292032203520382040 Assets2.43.24.04.75.35.75.95.64.93.61.70

Rational and Power FunctionsA rational function is simply the quotient of two polynomials.In general, a rational function has the form

where f(x) and g(x) are polynomial functions.Since the division by zero is not allowed, we conclude that the domain of a rational function is the set of all real numbers except the zeros of g (the roots of the equation g(x) = 0)

Rational and Power FunctionsExamples of rational functions:

Rational and Power FunctionsFunctions of the form

where r is any real number, are called power functions.We encountered examples of power functions earlier in our work. Examples of power functions:

Rational and Power FunctionsMany functions involve combinations of rational and power functions.Examples:

Applied Example: Driving CostsA study of driving costs based on a 2007 medium-sized sedan found the following average costs (car payments, gas, insurance, upkeep, and depreciation), measured in cents per mile:

A mathematical model giving the average cost in cents per mile is:

where x (in thousands) denotes the number of miles the car is driven in 1 year.

Miles/year, x500010,00015,00020,000Cost/mile, y ()83.862.952.247.1

Applied Example: Driving CostsBelow is the scatter plot associated with this data:51015202514012010080604020x (1000 miles/year)y ()C(x)


Applied Example: Driving CostsUsing this model, estimate the average cost of driving a 2007 medium-sized sedan 8,000 miles per year and 18,000 miles per year.SolutionThe average cost for driving a car 8,000 miles per year is

or approximately 68.8/mile.


Applied Example: Driving CostsUsing this model, estimate the average cost of driving a 2007 medium-sized sedan 8,000 miles per year and 18,000 miles per year.SolutionThe average cost for driving a car 18,000 miles per year is

or approximately 48.95/mile.


Constructing Mathematical ModelsSome mathematical models can be constructed using elementary geometric and algebraic arguments.

Guidelines for constructing mathematical models:Assign a letter to each variable mentioned in the problem. If appropriate, draw and label a figure.Find an expression for the quantity sought.Use the conditions given in the problem to write the quantity sought as a function f of one variable. Note any restrictions to be placed on the domain of f by the nature of the problem.

Applied Example: Enclosing an AreaThe owner of the Rancho Los Feliz has 3000 yards of fencing with which to enclose a rectangular piece of grazing land along the straight portion of a river.Fencing is not required along the river.Letting x denote the width of the rectangle, find a function f in the variable x giving the area of the grazing land if she uses all of the fencing.

Applied Example: Enclosing an AreaSolutionThis information was given:The area of the rectangular grazing land is A = xy.The amount of fencing is 2x + y which must equal 3000 (to use all the fencing), so:2x + y = 3000 Solving for y we get:y = 3000 2xSubstituting this value of y into the expression for A gives:A = x(3000 2x) = 3000x 2x2Finally, x and y represent distances, so they must be nonnegative, so x 0 and y = 3000 2x 0 (or x 1500). Thus, the required function is: f(x) = 3000x 2x2 (0 x 1500)

Applied Example: Charter-Flight RevenueIf exactly 200 people sign up for a charter flight, Leisure World Travel Agency charges $300 per person.However, if more than 200 people sign up for the flight (assume this is the case), then each fare is reduced by $1 for each additional person.Letting x denote the number of passengers above 200, find a function giving the revenue realized by the company.

Applied Example: Charter-Flight RevenueSolutionThis information was given.If there are x passengers above 200, then the number of passengers signing up for the flight is 200 + x.The fare will be (300 x) dollars per passenger.The revenue will beR= (200 + x)(300 x)= x2 + 100x + 60,000The quantities must be positive, so x 0 and 300 x 0 (or x 300). So the required function is: f(x) = x2 + 100x + 60,000 (0 x 300)

End of Chapter

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Functions and Their Graphs

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