We can classified function according to their properties or behavior under certain conditions. In mathematics we have even and odd function, rational function, polynomial function, injective function, bijective function, surjective function, composite function etc.

__Even and Odd function__:

Let f : A $\rightarrow$ B be a function. Then f is said to be even if for x $\in$ A we have

**f(-x) = f(x)** and is said to be odd if

**f(-x) = - f(x)**.

**Example 1:** Let f : R $\rightarrow$ R be a given function defined as f(x) = -x sin x, then if we

replace x by -x, then

f(-x) = -(-x) sin(-x)

= -x sin x

Hence the given function is even function.

**Example 2:**If f : R $\rightarrow$ R be a function and defined as f(x) = x^{3} cos x, then function

is odd function, since we replace x by -x, then

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

= -x^{3} cos x

**Polynomial Function**:

If a function define in the form of polynomial is known as polynomial function, like:

(a) f(x) = 3x^{4} - 4x^{2} + 1

(b) f(x) = x^{4} - x^{3} + x^{2} +x -1

If we have **f(x) = k**, then this type of function known as *constant *function.**Rational Function**:

If a function is in the form of f(x) = $\frac{g(x)}{h(x)}$, where g(x) and h(x) are the polynomial functions and

**h(x) $\neq$ 0**.

**Example:** f(x) = $\frac{x^2 + 2}{x - 1}$, where x$\neq$ 1, is a rational function.

**Inverse Function**:

Before we learn what is an Inverse function, lets learn some definitions.

**One to One function (Injective function)**

Let
f : A $\rightarrow$ B be a function. Then f is said to be one to one if
the distinct elements of A are mapped to the distinct elements of B.
That is no two elements in the set A has the same image. The A one to
one function is also known as the Injective function.

**Onto function (Surjective function)**

Let
f: A $\rightarrow$ B be a function. Then f is said to be Onto if every
element in B has a pre-image associated with it. A Onto function is
also known as the Surjective function.

In general in an Onto function, Co domain = Range.

**Bijective function**

A One to One and Onto function is called as the Bijective function.

Only Bijective functions will have inverse.

__Inverse function:__ Let f : A $\rightarrow$ B be a bijective function. Then the inverse function f ^{-1}: B $\rightarrow$ A is defined as f ^{-1}(y) = x, if and only if f(x) = y.

Let A ={a, b, c} and B ={1, 2, 3} be the two sets.

Then f : A $\rightarrow$ B defined by f = {(a, 1), (b, 2), (c, 3)} is a bijective since

the function maps every element to exactly one element and also all elements in

The inverse function of f is given by f ^{-1} = {(1, a), (2, b), (3, c)}

Consider the function y = 2x+3 defined on the set of all real numbers.

To find the inverse of this function, we first write x in terms of y, we have

Subtracting 3 on both sides we get y -3 = 2 x

Dividing by 2, we get x =$\frac{y-2}{3}$

Now replace x by f ^{-1}(x) and y by x

The required inverse function is

f ^{-1}(x) = $\frac{y-2}{3}$

**Composite function**:

Consider the functions f : A$\rightarrow$ B and g : B $\rightarrow$
C.

Let a $\in$ A . Then the image of a under f is f(a) and is
contained in B. Since the domain of g is B, the image of f(a) under g
is g(f(a)) which is in C. The rule which associates a $\in$ A to g( f
(a) ) $\in$ C is called composition of the function f and g.

**Definition**

If f : A $\rightarrow$ B and g : B $\rightarrow$ C are two functions then

gof : A $\rightarrow$ C is defined by gof(x) = g(f(x)).

**Example****:**

Let f : R$\rightarrow$ R and g: R$\rightarrow$ R are the two functions, where

f(x) = x +1 and g(x) = x^{2} + 1. Then calculate fog(x) and gof(x).

**Solution:**

Given f : R$\rightarrow$ R and g: R$\rightarrow$ R, where f(x) = x +1 and

g(x) = x^{2} + 1.

Now fog(x) = f{g(x)}

= f( x^{2} + 1)

= x^{2} + 1 + 1

= x^{2} + 2

Again gof(x) = g{f(x)}

= g(x + 1)

= (x + 1)^{2} + 1

= x^{2} + 2x +2

** Absolute Value Function**:

Absolute value function is a function, in which the image is always the absolute value of the real number. Absolute value function is called
so, because it omits the sign of the pre-image. Absolute value function
is the positive distance between two real numbers.

The graph of an absolute value function always lies in the positive region.__Lets consider an example:__
f(x) = |3x-5|

Put x = 2

Then f(x) = |3*2-5| = |1| = 1

Put x = 4

Then f(x) = |3*4-5| = |7| = 7

The points we have already found are the correct ones.

But unless we don't select points that give a negative value inside the absolute value, we won't get a clear picture of the graph.

So let consider two more values such that we get a negative value inside the absolute value.

Put x = -1

Then f(x) = |3*-1-5|

= |-8|

= 8

Put x = 0

Then f(x) = |3*0-5| = |-5| = 5

Now we plot the points.

The graph is given by:

**Monotonic Function**:

A function is known as monotonic function in an interval (a,b) if it is either decreasing or increasing on (a,b).

A function is said to be increasing on an interval (a,b) if the value of the function increase as the independent value increase,

i.e. if x1 > x

_{2} then f(x

_{1}) > f(x

_{2}).

Function is said to be decreasing on (a,b) if the value of the function decrease as the independent value increase, i.e. x1 > x

_{2} then f(x

_{1}) < f(x

_{2}).