1

Graphs and Functions

Session #3

Carlos da Maia, PhD

Business School

UNIVERSITY OF ST. THOMAS OF MOZAMBIQUE

3L5ECONS

Advanced Mathematics for Economics, September 2014

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Outline

1 Functions of One Variable

2 An Example of a Function

3 Domain of a Function

4 Graphs of Functions

5 Linear Functions

6 Linear Models

7 Non-linear Functions

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Introduction

Important in every area of pure and applied mathematics(including mathematics applied to economics);

The language of economic analysis is full of terms like demandand supply functions, cost functions, production functions,consumption functions, etc;

One variable is a function of another if this �rst variabledepends upon the second;

Example

The area of a circle is a function of its radius; if radius r is given,then the area A is determined;

4

Introduction

Important in every area of pure and applied mathematics(including mathematics applied to economics);

The language of economic analysis is full of terms like demandand supply functions, cost functions, production functions,consumption functions, etc;

One variable is a function of another if this �rst variabledepends upon the second;

Example

The area of a circle is a function of its radius; if radius r is given,then the area A is determined;

5

Introduction

Important in every area of pure and applied mathematics(including mathematics applied to economics);

The language of economic analysis is full of terms like demandand supply functions, cost functions, production functions,consumption functions, etc;

One variable is a function of another if this �rst variabledepends upon the second;

Example

The area of a circle is a function of its radius; if radius r is given,then the area A is determined;

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Introduction, cont.

We don't need a mathematical formula to show that onevariable is a function of another;

The table de�nes consumption expenditure as a function ofthe calendar year:

Example

Year 1998 1999 2000 2001 2002 2003

Consumption 5879.5 6282.5 6739.4 7055.0 7376.1 7760.9

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Introduction, cont.

We don't need a mathematical formula to show that onevariable is a function of another;

The table de�nes consumption expenditure as a function ofthe calendar year:

Example

Year 1998 1999 2000 2001 2002 2003

Consumption 5879.5 6282.5 6739.4 7055.0 7376.1 7760.9

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Introduction, cont.

The dependence between two variables can also be illustratedby means of a graph.

Figure: The La�er curve

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Introduction, cont.

Functions are given letter names such as f, g, or F;

If f is a function and x is a number in its domain D, then f(x) denotes thenumber that the function f assigns to x;

The symbol f(x) is pronounced "f of x", or often just "f x";

Note de di�erence:

f is a symbol forthe function; and

f(x) denotes the value of f at x.

If f is a function, we sometimes ley y denote the value of f at x, so

Example

y = f(x)

x is the independent variable or the argument of f;

y is the dependent variable, because the y (in general) depends on thevalue of x;

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Introduction, cont.

Functions are given letter names such as f, g, or F;

If f is a function and x is a number in its domain D, then f(x) denotes thenumber that the function f assigns to x;

The symbol f(x) is pronounced "f of x", or often just "f x";

Note de di�erence:

f is a symbol forthe function; and

f(x) denotes the value of f at x.

If f is a function, we sometimes ley y denote the value of f at x, so

Example

y = f(x)

x is the independent variable or the argument of f;

y is the dependent variable, because the y (in general) depends on thevalue of x;

11

Introduction, cont.

Functions are given letter names such as f, g, or F;

If f is a function and x is a number in its domain D, then f(x) denotes thenumber that the function f assigns to x;

The symbol f(x) is pronounced "f of x", or often just "f x";

Note de di�erence:

f is a symbol forthe function; and

f(x) denotes the value of f at x.

If f is a function, we sometimes ley y denote the value of f at x, so

Example

y = f(x)

x is the independent variable or the argument of f;

y is the dependent variable, because the y (in general) depends on thevalue of x;

12

Introduction, cont.

Functions are given letter names such as f, g, or F;

If f is a function and x is a number in its domain D, then f(x) denotes thenumber that the function f assigns to x;

The symbol f(x) is pronounced "f of x", or often just "f x";

Note de di�erence:

f is a symbol forthe function; and

f(x) denotes the value of f at x.

If f is a function, we sometimes ley y denote the value of f at x, so

Example

y = f(x)

x is the independent variable or the argument of f;

y is the dependent variable, because the y (in general) depends on thevalue of x;

13

Introduction, cont.

Functions are given letter names such as f, g, or F;

If f is a function and x is a number in its domain D, then f(x) denotes thenumber that the function f assigns to x;

The symbol f(x) is pronounced "f of x", or often just "f x";

Note de di�erence:

f is a symbol forthe function; and

f(x) denotes the value of f at x.

If f is a function, we sometimes ley y denote the value of f at x, so

Example

y = f(x)

x is the independent variable or the argument of f;

y is the dependent variable, because the y (in general) depends on thevalue of x;

14

Introduction, cont.

Functions are given letter names such as f, g, or F;

If f is a function and x is a number in its domain D, then f(x) denotes thenumber that the function f assigns to x;

The symbol f(x) is pronounced "f of x", or often just "f x";

Note de di�erence:

f is a symbol forthe function; and

f(x) denotes the value of f at x.

If f is a function, we sometimes ley y denote the value of f at x, so

Example

y = f(x)

x is the independent variable or the argument of f;

y is the dependent variable, because the y (in general) depends on thevalue of x;

15

Introduction, cont.

Functions are given letter names such as f, g, or F;

If f is a function and x is a number in its domain D, then f(x) denotes thenumber that the function f assigns to x;

The symbol f(x) is pronounced "f of x", or often just "f x";

Note de di�erence:

f is a symbol forthe function; and

f(x) denotes the value of f at x.

If f is a function, we sometimes ley y denote the value of f at x, so

Example

y = f(x)

x is the independent variable or the argument of f;

y is the dependent variable, because the y (in general) depends on thevalue of x;

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Introduction, cont.

The domain of the function f is the set of all possible values of theindependent variable;

The range is the set of corresponding values of the dependent varaible;

In economics, x is often called exogenous variable, whereas y is theendogenous variable;

A function is often de�ned by a formula such as: y = 2x2- 3x +8; thefunction is then the rule that assigns the number 2x2- 3x +8 to each value of x.

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Introduction, cont.

The domain of the function f is the set of all possible values of theindependent variable;

The range is the set of corresponding values of the dependent varaible;

In economics, x is often called exogenous variable, whereas y is theendogenous variable;

A function is often de�ned by a formula such as: y = 2x2- 3x +8; thefunction is then the rule that assigns the number 2x2- 3x +8 to each value of x.

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Introduction, cont.

The domain of the function f is the set of all possible values of theindependent variable;

The range is the set of corresponding values of the dependent varaible;

In economics, x is often called exogenous variable, whereas y is theendogenous variable;

A function is often de�ned by a formula such as: y = 2x2- 3x +8; thefunction is then the rule that assigns the number 2x2- 3x +8 to each value of x.

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Introduction, cont.

The domain of the function f is the set of all possible values of theindependent variable;

The range is the set of corresponding values of the dependent varaible;

In economics, x is often called exogenous variable, whereas y is theendogenous variable;

A function is often de�ned by a formula such as: y = 2x2- 3x +8; thefunction is then the rule that assigns the number 2x2- 3x +8 to each value of x.

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An Example of a Function

Example

The total dollar cost of producing x units of a product is given byC(x) = 100x

√x+500 for each nonnegative integer x. Find the cost

of producing:

1 16 units;

2 a units

3 Suppose the �rm produces a units; Find the increase in thecost from producing one additional unit.

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An Example of a Function, cont.

Example

Solution

1 C(16) = 100*16√16+500 = 100*16*4+500=6900

2 C(a) = 100a√a+500

3 The cost of producing a+1 units is C(a+1), so that theincrease in cost is C(a+1) - C(a) = 100(a+1)

√a+1+500 -

(100a√a+500) = 100[(a+1)

√a+1-a

√a]

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Domain of a Function

The de�nition of a function is incomplete unless its domain isspeci�ed;

The natural domain of the function f de�ned by f(x) = x3isthe set of all real numbers;

For te case of our example where C(x) = 100x√x+500

denotes the cost of producing x units of a product, the domainwas not speci�ed, but the natural domain is the set ofnumbers 0,1,3,..., n, where n is the maximum number of itemsthe �rm can produce.;

If output x is a continuous variable, the natural domain is theclosed interval [0, n].

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Domain of a Function

The de�nition of a function is incomplete unless its domain isspeci�ed;

The natural domain of the function f de�ned by f(x) = x3isthe set of all real numbers;

For te case of our example where C(x) = 100x√x+500

denotes the cost of producing x units of a product, the domainwas not speci�ed, but the natural domain is the set ofnumbers 0,1,3,..., n, where n is the maximum number of itemsthe �rm can produce.;

If output x is a continuous variable, the natural domain is theclosed interval [0, n].

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Domain of a Function

The de�nition of a function is incomplete unless its domain isspeci�ed;

The natural domain of the function f de�ned by f(x) = x3isthe set of all real numbers;

For te case of our example where C(x) = 100x√x+500

denotes the cost of producing x units of a product, the domainwas not speci�ed, but the natural domain is the set ofnumbers 0,1,3,..., n, where n is the maximum number of itemsthe �rm can produce.;

If output x is a continuous variable, the natural domain is theclosed interval [0, n].

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Domain of a Function, cont.

Examples

Find the domain of:

1 f(x)= 1x+3

2 g(x) =√2x+2

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Domain of a Function, cont.

Examples

Solution:

1 f(x)= 1x+3 ; x 6=−3

2 g(x) =√2x+2; [-1, ∞]

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Graphs of Functions

Figure: Some important graphs

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Graphs of Functions

Example

Consider the function f(x) = x2- 4x + 3. The values of f(x) forsome special choices of x are given in the following table.

Table: Values of f(x) = x2- 4x + 3

x 0 1 2 3 4

f(x) = x2- 4x + 3

Fill the table, then plot the points obtained from the table in axy-plane and, and then draw a smooth curve through this point.What is the name of the resulting graph?

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Graphs of Functions

Example

Find some of the points of the graph g(x) = 2x - 1, and sketch it.

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Linear functions

They occur often in economics and are de�ned as follows:

y = ax + b (a and b are constants)

The graph of the equation is a straight line.

a is called the slope of the function:

Proof.

f(x+1) - f(x) = a(x+1) + b - ax - b = aa measures the change in the value of the function when x increasesby 1 unit. For this reason, the number a is the slope of the line,and so is called the slope of the function.

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Linear functions, cont.

If a is positive, the line slants upward to the right, and thelarger the value of a, the steeper is the line;If a is negative, then the line slants downward to the right, andthe absolute value of a measures the steepness of the line.

Figure: Steepness of the line

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Linear functions, cont.The slope of a straight line

The slope of a straight line is:

a = y2−y1x2−x1 , x1 6=x2

where (x1, y1) and (x2, y2) are any two distinct points on thestraight line.

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Linear functions, cont.Exercises

Example

Determine the slopes of the 3 straight lines l, m, and n.

Figure: Determine the slopes

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Linear functions, cont.Exercises (Solutions)

Examples

al=3−24−1=

12

am=−2−21−2 = 4

an=−1−25−2 = -1

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Linear ModelsMost of the linear models in economics are approximations to more complicated models

Statistical methods have been devised to construct linearfunctions that approximate the actual data as closely aspossible

E

A UN report estimated that the European population was 641million in 1960, and 705 million in 1970. Use these estimates toconstruct a linear function of t that approximates the population inEurope (in millions), where t is the number of years from 1960(t=0 is 1960, t=1 is 1961, and so on). Then use the function toestimate the population in 1975, 2000, and 1930.

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Linear ModelsSolution

S

*If P denotes the population (in millions), we construct an equationof the form P = at + b;*We just need 2 points: (t1,P1) = (0,641) and (t2,P2) = (10, 705);*So P is our y and t our x;*From the point-slope formula we know that: in y = ax + b, a =y2−y1x2−x1 , x1 6=x2;

*So a = P2−P1

t2−t1 ,= 705−64110−0 =64/10;

*Therefore P = 64/10t + 641*What is P(15), P(40), and P(-30)?

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Linear ModelsOur estimates versus UN forecasts

Table: UN estimates versus our forecasts

Year 1930 1975 2000

t -30 15 40

UN estimates 573 728 854

Our forecasts 449 737 897

Our formula does not give very good results compared to UNestimates. To correct for this we need to go beyond linearfunctions.

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Quadratic Functions

To obtain acceptable descriptions of economic phenomenaeconomists often have to use more complicated functions;

Many economic models involve functions that either decreasedown to some minimum value and the increase, or elseincrease up to some maximum value and then decrease;

These are the quadractic functions:

Example

f(x) = ax2+ bx + c (a, b, and c are constants, a6=0)

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Quadratic Functions

In order to investigate the function f(x) = ax2+ bx + c in moredetail, we should �nd the answers to the following questions:

1 For which values of x is ax2+ bx + c = 0? → if and only if x

= −b±√b2−4ac2a →MEMORIZE THIS FORMULA

2 What are the coordinates of the maximum/minimum point P,also called the vertex of the parabola? → Use derivates

3 What are the coordinates of the maximum/minimum point P,also called the vertex of the parabola? → Use derivates

1 f(x) = ax2+ bx + c2 f'(x) = 2ax + b3 Equate it to zero → 2ax + b = 0;

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Quadratic Functions, cont.

if a>0, then f(x) = ax2+ bx + c has a minimum at x = -b/2a

if a>0 f(x) has a maximum at x = -b/2a

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Quadratic Functions, cont.Example

Example

The price P per unit obtained by a �rm in producing and selling Qunits is P = 102 - 2Q, and the cost of producing and selling Qunits is C = 2Q + 1/2Q2.(1) What is the expression of the pro�t?(2) What is the value of Q which maximizes pro�ts, and thecorresponding maximum pro�t? (Sol: 20 and 1000)

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Quadratic Functions, cont.Exercises

Determine the zeros and maximum/minimum points:

1 x2 + 4x

2 x2 + 6x + 18

3 -3x2 + 30x - 3

4 -x2 - 200x - 30000