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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

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Mathematics


Session

Functions, Limits and Continuity-1


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


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.


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.)

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

Find the domain of the following functions:


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

Find the range of the following functions:


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)

Y

f(x) = c

(0, c)

X

O

Domain = R

Range = {c}


Identity Function

Y

I(x) = x

450

X

O

Domain = R

Range = R


Modulus Function

Y

f(x) = x

f(x) = - x

X

O


Example

y = sinx

y = |sinx|


Greatest Integer Function

= greatest integer less than or equal to x.


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

defined by

Algebra of Real Functions


Algebra of Real Functions (Cont.)


Composition of Two Functions


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

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

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


Prove that is an even function.

Example – 5


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


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.)

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


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

Y

3

X

O

x

x takes the values 3.1

3.002

3.000005

……..

3.00000000000257 etc.


Existence Theorem on Limits


Example – 7

Which of the following limits exist:


Example - 7 (ii)


Properties of Limits

If and

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


Algebraic Limits (Substitution Method)

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


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

takes the form

Algebraic Limits (Factorization Method)


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

takes the form

Algebraic Limits (Rationalization Method)


Standard Result

If n is any rational number, then


Example – 8 (i)


Example – 8 (ii)


Example – 8 (iii)


Solution Cont.


Example – 8 (iv)


Example – 8 (v)


Thank you


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