Economics 2301

Lecture 5

Introduction to Functions Cont.

Necessary & Sufficient Conditions

Symbol means “implies” PQ means “if P then Q,” “P implies

Q,” “P only if Q.” P is a sufficient condition for Q, for if P holds then if follows that Q holds.

This expression also shows that Q is a necessary condition for P cause P can only hold if Q holds.

Example of Necessary and Sufficient Conditions

f strictly concave f concave

If and only if

BABA xfxfxxff 11 concave is

Power Functions

funciton. theofexponent theis parameter The

constants.areand

)(

form general theakesfunction tpower A

p

pk

kxxf P

Rules of Exponents

a

a

a

aaa

abba

bab

a

baba

mnn mnm

pp

y

x

y

x

xyyx

xx

xx

x

xxx

xxx

xx

xx

x

)(

1

1

/

1

0

Power functions with p equal even integer

F(0)=0 If k>0, then f(x) reaches a global

minimum at x=0. If k<0, then f(x) reaches global maximum at x=0.

These functions are symmetric about the vertical axis. They are strictly convex if k>0 or strictly concave if k<0.

Figure 2.13 Power Functions in Which the Exponent Is a Positive Even Number

Power function when p is positive odd integer

If k>0, then the function is monotonic and increasing. If k<0, then the function is monotonic and decreasing.If p≠1 and k>0, the function is strictly concave for x<0 and strictly convex for x>0.If p≠1 and k<0, the function is strictly convex for x<0 and strictly concave for x<0.

Figure 2.14 Power Functions in Which the Exponent Is a Positive Odd Integer

Power function with p negative integer

The function is non-monotonicThe function is not continuous and has a vertical asymptote at x=0.

Power function with p negative integer

p

x

p

x

p

x

p

x

p

x

p

x

kxandkx

kxandkx

kxandkx

limlim

limlim

limlim

00

00

00

theninteger, odd negative a is p and 0k If

theninteger,even negative a is p and 0k If

theninteger,even negative a is p and 0k If

Power function with p negative integer

0.for x concavestrictly and 0for xconvext strictly

is then integer, odd negative a is p and 0k If

0.for xor 0for xconvex strictly

is then integer,even negative a is p and 0k If

theninteger, odd negative a is p and 0k If

limlim00

p

p

p

x

p

x

kx

kx

kxandkx

Figure 2.15 Power Functions in Which the Exponent Is a Negative Integer

Polynomial Functions

. to1

from integers are polynomial in the exponents theand

numbers real are ,,2,1,0, parameters the

)(

form theakesfunction t polynomial univariateA 2

210

n

nia

xaxaxaaxfy

i

nn

Polynomial Functions

The degree of the polynomial is the value taken by the highest exponent.A linear function is polynomial of degree 1.A polynomial of degree 2 is called a quadratic function.A polynomial of degree 3 is called a cubic function.

Roots of Polynomial Function

a

acbbxx

cbxaxy

bax

bxay

2

4, :roots

:function Quadratic

/ :root

:functionLinear

zero. equalfunction themakethat

argument its of values theare polynomial a of roots The

2

21

2

3 cases for roots of quadratic function

b2-4ac>0, two distinct roots.b2-4ac=0, two equal rootsb2-4ac<0, two complex roots

Quadratic example

34

12

4

57

4

257

4

24497

2*2

2*3*477

2

4

2

1

4

2

4

57

4

257

4

24497

2*2

2*3*477

2

4

372

22

2

22

1

2

a

acbbx

a

acbbx

xxy

Plot of our Quadratic function

Roots of Quadratic Equation

-4

-2

0

2

4

6

8

10

12

14

-2 -1 0 1 2 3 4 5

x

y y

Exponential Functions

The argument of an exponential function appears as an exponent.Y=f(x)=kbx

k is a constant and b, called the base, is a positive number.f(0)=kb0=k

Exponential Functions

0

and with decreasesly montonical 0,b1When

axis.-y theacross function theof

reflection a is 1

ofgraph theThus1

b

1

exponents of rulesour Using

0

case In this . with increases then ,1When

x.of any valuefor parameter the

ofsign theas same theisfunction thisofsign The

lim

lim

x

x

x

x

x

xx

x

x

x

x

kb

xkb

b

bb

b

kb

xbb

k

Figure 2.16 Some Exponential Functions

Exponential functions with k<0

Exponential Functions with k<0, b=3/2 and 2/3

-8000

-7000

-6000

-5000

-4000

-3000

-2000

-1000

0

1000

-25 -20 -15 -10 -5 0 5 10 15 20 25

x

y

y2

y1