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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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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]

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

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

### 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 - 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.

### Example – 7

Which of the following limits exist:

### 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

### Standard Result

If n is any rational number, then

Thank you