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# Mathematics - PowerPoint PPT Presentation

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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Functions, Limits and Continuity-1

• 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 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.

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).

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 . . . }

Find the domain of the following functions:

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}

Find the range of the following functions:

• -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]

Y

f(x) = c

(0, c)

X

O

Domain = R

Range = {c}

Y

I(x) = x

450

X

O

Domain = R

Range = R

Y

f(x) = x

f(x) = - x

X

O

Example is

y = sinx

y = |sinx|

= greatest integer less than or equal to x.

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

defined by

Algebra of Real Functions

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

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 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

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 is

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

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

Y

x takes the values

2.91

2.95

2.9991

..

2.9999 ……. 9221 etc.

x

X

O

3

Y

3

X

O

x

x takes the values 3.1

3.002

3.000005

……..

3.00000000000257 etc.

Which of the following limits exist:

If and

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

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

takes the form

Algebraic Limits (Factorization Method)

takes the form

Algebraic Limits (Rationalization Method)

If n is any rational number, then