Grade Level: HG1

Course: Pre calculus

Branch: Functions

Topic: Identity Functions , Odd and Even functions

Identity Function

For any set A, the identity function I : A � A is the function that takes an

element to itself; in other words, for every element x belongs to A , I(x) = x .

Properties

The identity function is an Injective ( one-one function) and Subjective ( Onto

Function )

Understanding the Identity Function

(i) The identity function on a set A is the function that does nothing to each

element of A.

(ii) The identity function on R is the familiar function defined byf(x) =x . Its

graph in the plane is the diagonal 450 line from lower left to upper right

through the origin.

Example

From the knowledge of coordinate geometry, y=x represents a straight line

passing through the origin and inclined at angle 45° with the x axis.

Clearly the domain and range of the identity function are both equal to R.

EVEN AND ODD FUNCTIONS

Even Functions

Let f(x) be a real-valued function of a real variable. Then f is even if the

following equation holds for all x in the domain of f

f(-x) = f(x)

Geometrically, the graph of an even function is symmetric with respect to the y-

axis, meaning that its graph remains unchanged after reflection about the y-axis.

Examples of Even functions

1. f(x) = |x|

2. f(x) = x2

3. f(x) = cos(x)

Odd Functions

Let f(x) be a real-valued function of a real variable. Then f is odd if the following

equation holds for all x in the domain of f

f(-x) = -f(x)

Geometrically, the graph of an odd function has rotational symmetry with

respect to the origin, meaning that its graph remains unchanged after rotation of

180 degrees about the origin.

Examples of Odd functions

1. f(x) = 1/x

2 . f(x) = x3

3 . f(x) = sin(x)

Basic Properties of Even & Odd functions

• The only function which is both even and odd is the constant function

which is identically zero (i.e., f(x) = 0 for all x).

• The sum of an even and odd function is neither even nor odd, unless one

of the functions is identically zero.

• The sum of two even functions is even, and any constant multiple of an

even function is even.

• The sum of two odd functions is odd, and any constant multiple of an odd

function is odd.

• The product of two even functions is an even function.

• The product of two odd functions is an even function.

• The product of an even function and an odd function is an odd function.

• The quotient of two even functions is an even function.

• The quotient of two odd functions is an even function.

• The quotient of an even function and an odd function is an odd function.

QUIZ

1 . Choose the graph of odd function

A ) Graph A

B ) Graph B

C ) Graph C

D ) Graph B & C

2 . Verify whether the function f (x) = (x - 2)2 is even, odd or neither?

A ) Even

B ) Odd

C ) Neither

D ) Both Even and odd

3. Verify whether the function h (x) = |x| - 2 is even, odd or neither?

A ) Even

B ) Odd

C ) Neither

D ) Both Even and Odd

4. Verify whether the function g (x) = x2 + 4 is even, odd or neither?

A ) Even

B ) Odd

C ) Neither

D ) Both Even and Odd

5. Verify whether the function p (x) = 2x is even, odd or neither?

A ) Even

B ) Odd

C ) Neither

D ) Both Even & Odd

6 . Verify whether the function r(x) = 2x3 – 5x is even, odd or neither?

A ) Even

B ) Odd

C ) Neither

D ) Both Even & Odd

7 . which of the following graphs represents Identity function

A ) Graph A

B ) Graph B

C ) Graph C

D ) None of these