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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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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

FunctionIf 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 RangeThe 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.) is

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 is

Find the domain of the following functions:

Example– 1 (ii) is

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 is

Find the range of the following functions:

Example – 2(ii) is

- -1 cos2x 1 for all xR
- -3 3cos2x 3 for all xR
- -2 1 + 3cos2x 4 for all xR
- -2 f(x) 4
- Hence , range of f = [-2, 4]

Greatest Integer Function is

= 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 FunctionsExample - 3 is

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 is

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 is

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. is

Example – 5Example - 6 is

Let the function f be f(x) = x3 - kx2 + 2x, xR, 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 xR

-x3 - kx2 - 2x = - x3 + kx2 - 2x for all xR

- 2kx2 = 0 for all xR
- 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 FunctionAs x approaches 3 from left hand side of the number line, f(x) increases and becomes close to 6

Limit of a Function (Cont.) is

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

Example – 7 is

Which of the following limits exist:

Algebraic Limits (Substitution Method) is

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 it

If n is any rational number, then

Thank you it

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