Slide1 l.jpg
This presentation is the property of its rightful owner.
Sponsored Links
1 / 42

Mathematics PowerPoint PPT Presentation


  • 84 Views
  • Uploaded on
  • Presentation posted in: General

Mathematics. Session. Functions, Limits and Continuity-1. Session Objectives. Function Domain and Range Some Standard Real Functions Algebra of Real Functions Even and Odd Functions Limit of a Function; Left Hand and Right Hand Limit

Download Presentation

Mathematics

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Slide1 l.jpg

Mathematics


Session l.jpg

Session

Functions, Limits and Continuity-1


Session objectives l.jpg

Session Objectives

  • Function

  • Domain and Range

  • Some StandardReal Functions

  • Algebraof Real Functions

  • Even and Odd Functions

  • Limit of a Function; Left Hand and Right Hand Limit

  • Algebraic Limits : Substitution Method, Factorisation Method, Rationalization Method

  • Standard Result


Function l.jpg

If f is a function from a set A to a set B, we represent it by

If f associates then we say that y is the image of the

element x under the function or mapping and we write

Function

If A and B are two non-empty sets, then a rule which associates

each element of A with a unique element of B is called a function

from a set A to a set B.

Real Functions: Functions whose co-domain, is a subset of R

are called real functions.


Domain and range l.jpg

The set A is called the domain of the function and the set B is

called co-domain.

Domain and Range

The set of the images of all the elements under the mapping

or function f is called the range of the function f and represented

by f(A).


Domain and range cont l.jpg

Domain and Range (Cont.)

For example: Consider a function f from the set of natural

numbers N to the set of natural numbers N

i.e. f : N N given by f(x) = x2

Domain is the set N itself as the function is defined for all values of N.

Range is the set of squares of all natural numbers.

Range = {1, 4, 9, 16 . . . }


Example 1 l.jpg

Example– 1

Find the domain of the following functions:


Example 1 ii l.jpg

Example– 1 (ii)

The function f(x) is not defined for the values of x for which the

denominator becomes zero

Hence, domain of f = R – {1, 2}


Example 2 l.jpg

Example- 2

Find the range of the following functions:


Example 2 ii l.jpg

Example – 2(ii)

  • -1  cos2x  1 for all xR

  • -3  3cos2x  3 for all xR

  • -2  1 + 3cos2x  4 for all xR

  •  -2 f(x)  4

  • Hence , range of f = [-2, 4]


Some standard real functions constant function l.jpg

Some Standard Real Functions (Constant Function)

Y

f(x) = c

(0, c)

X

O

Domain = R

Range = {c}


Identity function l.jpg

Identity Function

Y

I(x) = x

450

X

O

Domain = R

Range = R


Modulus function l.jpg

Modulus Function

Y

f(x) = x

f(x) = - x

X

O


Example l.jpg

Example

y = sinx

y = |sinx|


Greatest integer function l.jpg

Greatest Integer Function

= greatest integer less than or equal to x.


Algebra of real functions l.jpg

Multiplication by a scalar: For any real number k, the function kf is

defined by

Algebra of Real Functions


Algebra of real functions cont l.jpg

Algebra of Real Functions (Cont.)


Composition of two functions l.jpg

Composition of Two Functions


Example 3 l.jpg

Example - 3

Let f : R R+ such that f(x) = ex and g(x) : R+R such that g(x) = log x, then find

(i) (f+g)(1) (ii) (fg)(1)

(iii) (3f)(1) (iv) (fog)(1) (v) (gof)(1)

Solution :

(i) (f+g)(1) (ii) (fg)(1) (iii) (3f)(1)

= f(1) + g(1) =f(1)g(1) =3 f(1)

= e1 + log(1) =e1log(1) =3 e1

= e + 0 = e x 0 =3 e

= e = 0

(iv) (fog)(1) (v) (gof)(1)

= f(g(1)) = g(f(1))

= f(log1) = g(e1)

= f(0) = g(e)

= e0 = log(e)

=1 = 1


Example 4 l.jpg

Example – 4

Find fog and gof if f : R  R such that f(x) = [x] and g : R  [-1, 1] such that g(x) = sinx.

Solution:We have f(x)= [x] and g(x) = sinx

fog(x) = f(g(x)) = f(sinx) = [sin x]

gof(x) = g(f(x)) = g ([x]) = sin [x]


Even and odd functions l.jpg

Even and Odd Functions

Even Function : If f(-x) = f(x) for all x, then

f(x) is called an even function.

Example: f(x)= cosx

Odd Function : If f(-x)= - f(x) for all x, then

f(x) is called an odd function.

Example: f(x)= sinx


Example 5 l.jpg

Prove that is an even function.

Example – 5


Example 6 l.jpg

Example - 6

Let the function f be f(x) = x3 - kx2 + 2x, xR, then

find k such that f is an odd function.

Solution:

The function f would be an odd function if f(-x) = - f(x)

 (- x)3 - k(- x)2 + 2(- x) = - (x3 - kx2 + 2x) for all xR

 -x3 - kx2 - 2x = - x3 + kx2 - 2x for all xR

  • 2kx2 = 0 for all xR

  • k = 0


Limit of a function l.jpg

x

2.5

2.6

2.7

2.8

2.9

2.99

3.01

3.1

3.2

3.3

3.4

3.5

f(x)

5.5

5.6

5.7

5.8

5.9

5.99

6.01

6.1

6.2

6.3

6.4

6.5

Limit of a Function

As x approaches 3 from left hand side of the number line, f(x) increases and becomes close to 6


Limit of a function cont l.jpg

Limit of a Function (Cont.)

Similarly, as x approaches 3 from right hand side of the number line, f(x) decreases and becomes close to 6


Left hand limit l.jpg

Left Hand Limit

Y

x takes the values

2.91

2.95

2.9991

..

2.9999 ……. 9221 etc.

x

X

O

3


Right hand limit l.jpg

Right Hand Limit

Y

3

X

O

x

x takes the values 3.1

3.002

3.000005

……..

3.00000000000257 etc.


Existence theorem on limits l.jpg

Existence Theorem on Limits


Example 7 l.jpg

Example – 7

Which of the following limits exist:


Example 7 ii l.jpg

Example - 7 (ii)


Properties of limits l.jpg

Properties of Limits

If and

where ‘m’ and ‘n’ are real and finite then


Algebraic limits substitution method l.jpg

Algebraic Limits (Substitution Method)

The limit can be found directly by substituting the value of x.


Algebraic limits factorization method l.jpg

When we substitute the value of x in the rational expression it

takes the form

Algebraic Limits (Factorization Method)


Algebraic limits rationalization method l.jpg

When we substitute the value of x in the rational expression it

takes the form

Algebraic Limits (Rationalization Method)


Standard result l.jpg

Standard Result

If n is any rational number, then


Example 8 i l.jpg

Example – 8 (i)


Example 8 ii l.jpg

Example – 8 (ii)


Example 8 iii l.jpg

Example – 8 (iii)


Solution cont l.jpg

Solution Cont.


Example 8 iv l.jpg

Example – 8 (iv)


Example 8 v l.jpg

Example – 8 (v)


Slide42 l.jpg

Thank you


  • Login