A function is an expression involving one or more variables. It is a rule that relates set of inputs with some outputs.

For example, if y is a function in terms of x.

y = f(x)

By putting different values of x, we get different values of y.

## Classification of function

A function is classified into four types.

### Even function

The function is said to be even if we put -x in the place of x and there is no change in the value of a function. It remains the same and we get back the same function.

**Formula**

Suppose we have a function f(x) then put -x in the place of x and we get back the same function.

f(-x)=f(x)

### Odd function

The function is said to be an odd function if we put -x in the place of x and we get the same function but with the opposite sign.

**Formula**

Suppose we have a function f(x) then put -x in the place of x and we get back the same function but the sign is reversed.

f(-x)=-f(x)

**For example**, we have a function

f(x)=x^{3}

Then put negative sign

f(-x)=(-x)^{3}

We get

f(-x)=-x^{3}

It means

f(x)=-f(x)

Put different values of x for plotting a graph.

**Graph**

The graph of an odd function is symmetric about the origin.

## Neither even nor odd function

The function is said to be neither even nor odd if it does not obey the even and odd function formula.

**Formula**

f(-x) ≠ f(x)

f(-x) ≠ -f(x)

**For example, **we have a function

f(x)=2x^{5}+3x^{2}+3

Then put negative sign

f(-x)=2(-x)^{5}+3(-x)^{2}+3

We get

f(-x)=-2x^{5}+3x^{2}+3

f(x)=-(2x^{5}-3x^{2}-3)

It means

f(-x) ≠ f(x) and f(-x) ≠ -f(x)

Put different values of x for plotting a graph.

**Graph**

The graph is neither symmetrical about the y-axis nor its origin.

## Both even and odd function

Some functions are both even and odd.

**Formula**

f(-x) =f(x)

f(-x)= -f(x)

**For example**

The only function that is even and odd is f(x)=0

**Graph **

The graph is symmetrical about both the origin and y-axis.

## Explanation of an even function

**Example -1 **

Let we have an example

f(x)=x^{2}

Put -x in place of x

f(-x)=(-x)^{2}

f(-x)=x^{2}

It means

f(x)=f(-x)

So x^{2} is an even function.

**Graphical representation **

Put a different value of x for plotting a graph.

The graph is symmetric about the y-axis.

Similarly x^{4},x^{6},x^{8} and x^{10} are even functions.

From the above example it is concluded that the function which contains an even exponent may be an even function but not always. For example (x-1)^{2} is not an even function.

**Example-2 **

Let’s have a trigonometric function

f(x)=cosx

Put -x in the place of x

f(-x)=cos(-x)

As cosine absorb negative sign

f(-x)= cosx

It means

f(-x)=f(x)

So cosx is also an even function.

**Example-3**

Let

f(x)=4x^{4}-7x^{2}

Put -x in the place of x

f(-x)=4(-x)^{4}-7(-x)^{2}

f(-x)=4x^{4}-7x^{2}

It means

f(-x)=f(x)

So 4x^{4}-7x^{2} is an even function.

**Example-4**

Let

f(x)=7x^{4}-x^{14}

Put -x in the place of x

f(-x)=7(-x)^{4}-(-x)^{14}

f(-x)=7x^{4}-x^{14}

It means

f(-x)=f(x)

So 7x^{4}-x^{14} is an even function.

**Example-5**

Let

f(x)=5cosx-sin^{2}x

Put -x in the place of x

f(-x)=5cos(-x)-sin^{2}(-x)

f(-x)=5cosx-sin^{2}x

It means

f(-x)=f(x)

So 5cosx-sin^{2}x is an even function.

## Properties of an even function

**Sum and differences**

The sum of two even functions is an even function.

The difference of two even functions is also an even function.

**Product and quotient **

The product of two even functions is an even function.

The quotient of two even functions is an even function.

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

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

**In terms of composition**

The composition of two even functions is an even function.

The composition of even and odd functions is an even function.

## Concepts Berg

**What Is an Even Function?**

The function is said to be an even function if f(-x)=f(x) means if we put -x in the place of x in the given function, the function remains the same.

For example f(x)=x^{2} is an even function.

It means the function is positive for both +ve x-axis and -ve x-axis.

The graph of the even function is symmetric about the y-axis.

**What does an odd function look like?**

The function is said to be an odd function if f(-x)=-f(x) means if we put -x in the place of x in the given function, we get back the same function but with the opposite sign.

For example f(x)=x^{3} is an odd function.

The graph of odd functions looks symmetric about origin.

**Do you know any function which is neither odd or even?**

The function is said to be neither even nor odd if it does not obey the even and odd function formula.

For example a function f(x)=x^{5}+2x^{3}+1 is neither even nor odd.

**Why is sine an odd function?**

Sine is said to be an odd function because it obeys an odd function formula. As

f(x)=sinx then we put -x in the place of x, we get

f(-x)=sin(-x)

f(-x)=-sinx

It means f-(x)=-f(x) so it is an odd function.

**What are the properties of even functions?**

a) The sum of two even functions is an even function.

b)The difference of two even functions is also an even function.

c)The product of two even functions is an even function.

d)The quotient of two even functions is an even function.

e)The composition of two even functions is an even function.

f)The composition of even and odd functions is an even function.

**Is cosine an even function?**

As cosine obey the formula of an even function when put -ve sign in the function like

f(x)=cosx

f(-x)=cos(-x)

f(-x)=cosx

It means f(-x)=f(x). So it is an even function.

**Is y=4 an even function?**

For the function we must have sets of some inputs and outputs and it involves one or more variables. y=4 is not an even function because it does not fulfill the requirements of even function. It is just an even number.

- Is y=1 even or odd function?
- Why is f(x) =5 an even function?
- What are 2 daily life examples of even and odd functions each?
- Is the product of an even and an odd function odd or even?

The product of an even and an odd function is an odd function. For example f(x)=x^{2 } is an even function and g(x)=x^{3} is an odd function.

It means f(-x)=f(x) and g(-x)=-g(x).

So f-(x).g(-x)=f(x)g(-x) which is equal to -f(x).g(x). So it is an odd function.

**Is x^2 sin nx even or odd?**

According to definition of an odd function f(-x)=-f(x)

Let f(x)=x^{2} sin nx

f(-x)= (-x)^{2}sin n(-x)

f(-x)=-x^{2}sin nx which is equal to -f(x).

So it is an odd function.

**How do you determine if a function is odd or even with absolute value?**

The absolute value is f(x)=IxI.

The f(x) is x if x>0 and f(x)is -x if x<0. In absolute value function f(-x)=f(x) so it is an even function.

**Are even functions usually also symmetric?**

The graph of an even function is symmetric about the y-axis.