session 01 - functions

7/29/2019 Session 01 - Functions

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Session 01

Functions


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A relation, mapping between a set of

inputs and a set of permissible outputswith the property that each input is

related to exactly one output

FUNCTIONS

Demand being a function of price

Position of a moving particle as being a function of time

Definition

A, B two non empty sets.

Each element x in A maps to unique element in y in B

atuv

A

X

B

Y

A

X

B

Y

Z


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Note (Concerning the definition)

I The rule need not involve and explicit formula.

II A, B need not be subsets of

f(1)=2 f(2)=2 f(3)=15

BA

HT

1/4

TT

TH

HH

function: probability of an event in A

P(HH)=1/4 P(HT)=1/4 P(TH)=1/4 P(TT)=1/4As coordinates of f:

(1,2),(2,2),(3,15) As coordinates of f:

(HH, 1/4),(HT, 1/4),(TH, 1/4),(TT, 1/4)


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For x=2, y=4

For x=1/2, y=1/4

;2xy x

Independent Variable

from the set Dependent vari

ablecorresponding to given

X value from

;2xy x

Independent Variable

fromDependent vari

able

corresponding to given

X from

X can take only

integers.

X cannot take .

Domain:

If f is a given function, the collection of all objects x whichare first coordinates of pairs belonging to f is defined the

domain of f and denoted by Df


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Range of a function

The collection of all objects y which are second co-ordinates of

pairs belonging to f is defined the range off

fy : y f x ,x D Range of f =

x1

x2

x3 y4

y2

y3

y1 Range

Domain

Co

domain


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We write, y = f(x) when ever the pair (x,y) belongs to f


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Constant function

In Mathematics, a constant function is a function whose values

do not vary and thus are constant.

Ex: f(x) = 4


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Identity function

Always returns the same value that was used as its argument.

f(x) = x

Translation function

A translation function is a function T, which sends every real number

x into x + 1, that is T(x) = x + 1


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Graphical representation of a functions

x x1 x2 x3 x4 ... xn

y y1 y2 y3 y4 ... yn


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224)(,4)( xxgxxf

When we look at the two functions, we should identify that

24 x 0,

and 24 x 0

And by taking the square 2 2y 4 x 2 2x y 4;

This gives the locus of a circle with radius 2 and center at(0,0)

Therefore the two

graphs are

* When we consider the domain from -3 to -2,

functions gives complex numbers and cannot

represent on a real line.


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1y

x

That is xy = 1 (where x 0 & y 0 )

With the domain of all nonzero numbers

y is small when x is large and y is large when x is small

x and y are either both positive a both negative

At x=0, function is not defined

x y-2 -0.5

-1 -1

-0.5 -2

1 1

2 0.5


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One-to-one functions/ injective functions

For one value in the domain, there exists only one vale in the range

A function y = f(x) is called an one-to-one function if for each from

the range of f there exists exactly one x in the domain of f which is

related to y.Suppose f is a function such that 1 2 fx ,x D

1 2 1 2f(x ) = f(x ) x = x


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If there are two values in the range are equal, then the corresponding

values in the domain should be equal.

Range is all f(x), where xs are from the domain.

Then if f(x1)=f(x2)=y and x1x2,

then both x1 and x2 are mapped to one value y.

That is function f is not one to one.

Ex: f(x)= 2y 9 x

Suppose f(x1) = f(x2) ; 9-x12 = 9-x22 ; x12 = x22

that is x1= x2, that is x1x2 , that is f(x) is not one to one.


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On to functions / surjective

A function f from A to B is called onto if for all b in B there is an

a in A such that f(a) = b. All elements in B are used

That is range of f is same as B

Definition (bijection)

A function is called a bisjection, if it is onto and one-to-one.

Every bijection has a function called the inverse function.


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Inverse Functions

f f-1

A B A B

We have a function from A to B.

We have to find a function from B to A.

if every element y of B, is uniquely mapped to an element

x in A.


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Let f be a bisection from a set A to a set B.

f is onto

For one value in the domain, there

exists only one vale in the range

For all b in B there is an a in A

such that f(a) = b

for every element b of B, g(b) = a,

where f(a) = b.

Then the function g is called the inverse function of f,

and it is denoted by f-1,

such an a is unique for each b in B

f is one to one


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a1

a2

a3

b1

b2

b3

f f-1

if f is not onto?

if f is not one to one? 1y f x x f y

b1= f(a1) a1=f-1 (b1)


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Graphs of Inverse Functions

The reflection of the point (a,b) about the line y = x is the point

(b,a)


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Exponents

The exponent of a number says how many times to use the number

in a multiplication.

In 82 the "2" says to use 8 twice in a multiplication,

so 82 = 8 8 = 64

Exponents are also called Powers or Indices.


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Exponential Functions

In mathematics, the exponential function is the function ex,

where e is the number (approximately 2.718281828 and called asEulers number )

Sometimes the term exponential function is used more

generally for functions of the form y=ax, where the base a is any

positive real number, not necessarily e.

Find the graph ofy=ax

Graph of y=ex


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The logarithm of a number is the exponent by which anotherfixed value, the base, must be raised to produce that number.

For example, the logarithm of 1000 to base 10 is 3, because 1000

is 10 to the power 3: 1000 = 10 10 10 = 103.

More generally, ifx= by, then yis the logarithm ofxto base b,

and is written y= logb(x), so log10(1000) = 3.

logarithm


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Logarithmic Functions

If a is any positive number other than 1, then f(x) = ax

, theexponential function with base a, is one-to-one, and hence has an

inverse.

The logarithmic function with base a, written loga(x), is the

inverse of the exponential function ax.

The Natural Logarithm

The logarithm with base e is called the natural logarithm, and

it is denoted ln.

Natural Logarithm of x = ln x = loge x


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It can be shown from the series definitionsthat the sine and cosine functions

are the imaginary and real parts, respectively, of the complex exponential

function when its argument is purely imaginary:

This identity is called Euler's formula.

Then,


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Hyperbolic function

In mathematics, hyperbolic functions are analogs of the ordinary

trigonometric, or circular, functions. The basic hyperbolic functions are the

hyperbolic sine "sinh" and the hyperbolic cosine "cosh" ,from which are

derived the hyperbolic tangent "tanh"


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Consider a circle with a unit radius, points (cos t, sin t) form a circle with a unit radius

Consider a hyperbola with a unit radius

the points (cosh t, sinh t) form the right half of

the equilateral hyperbola

session 01 - functions

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