Types of Functions

The term ‘function’ was originated in year 1673 by Gottfried Lebiniz. Functions are the equations that relate an output with an input. The value of the output depends on the value of input. A function is of the form y = f(x) where for every value of x, there is only one value of y. For y = f(x), x is the independent variable whose value is not dependent on any other value and ‘y’ is a dependent variable whose value depends on value of ‘x’. For example: - if we have f(x) = x + 1, we can choose any random value of x and for every value of x, there is a different value of ‘y’.

The input and the output of a function make an ordered pair and is written as like this: -
(Input, Output)
Therefore a function is sometimes called a set of ordered pairs. One input produces only one output of the function. Output is always dependent on the value of the input which is always independent.

Types of functions

There are many types of functions, the important ones are listed below: -
1. Explicit functions: A function is said to be an explicit function if the dependent variable is fully expressed in terms of independent variable. For example: - f(x) = x^2 + 3x – 5 is an explicit function because their relation is such that value of f(x) depends on value of ‘x’.
2. Implicit functions: A function is said to be an implicit function if the dependent variable is not expressed in terms of independent variable. For example: - 5x^2 + 5y^2 = 25 is an implicit function because their relation is not in that term in which we can say that y is dependent on ‘x’.
3. Injective functions: A function is said to be injective when every element of its domain is mapped to at most one element of its co domain. These are also called one to one function as one element is linked to only one other element and one to many is not possible. That means for every value of ‘f’ there is only one value of ‘x’ that corresponds to ‘f’.
For example: - f(x) = x + 8 is one to one as for every possible value of f(x) there is one value of x.
4. Bijective functions: A function is said to be bijective when every element of its co domain is mapped to one elemnt of its domain. These are also called onto functions and one to one functions.

Learn types functions with examples

What is a Function?
The definition of function is as follows:
“Given two sets, an input set X and an output set Y, a function f from X to Y (f: X->Y) is a set of ordered pairs (x, y) formed using the function f. It is formed in such a way that for every x (input number) in domain X there is a unique y (output number) in co-domain Y. “F” represents the set of all ordered pairs so formed.”

Domain, Co-domain and Range of a Function
Domain is the set of all possible inputs to the function f and Co-domain is the set of all possible outputs to the function f. The set of all outputs that match the inputs in the domain as per the function f form the range or image of the function calculated by the equation y = f(x).  So, the image or range is usually a subset of co-domain.

Function Example
An example of a function will clearly answer the question: What is a function?
Function:  f(x) = 2x where x: R -> {1, 2}. “R” represents that x is a real number
From this we can calculate the output as shown below:
f(1) =  2(1) = 2
f(2) = 2(2) = 4

Thus, F = {(1, 2), (2, 4)}
Domain = 1, 2
Range = 2, 4
               
Types of Function
The different types of functions are:
• Injections or One-to-One Function: The one-to-one function is based on the property that:  “a” must equal “b” if f(a) is equal to f(b) denoted as f(a) = f(b).

• Surjections or Onto Function: In case of onto function, for each and every number y in the co-domain Y, there is an input x in Domain X formed by the function f where f(x) = y.

• Bijections: A function is said to be a bijective function if it is both one-to-one and onto.

• Inverse function: A function that is formed from co-domain to domain i.e., f: Y->X is termed as Inverse function. For instance, conversion of temperature from centigrade to fahrenheit and vice versa is an example of inverse function.

• Identity function: When every input x in domain X maps to itself, it is called Identity function. It is represented by the function y=f(x) = x. In such case X=Y. For example, f (1) = 1, f(2) = 2.

Derivative of Tan and Tan-1

Derivative of Tan:
We know that in Trignometric functions, tan x = sinx/cosx
Let us derive tan x, with respect to x
Here, f(x) = sin x and g(x) = cos x
Using  Quotient Rule for Derivatives,



 d/dx  [f(x)/g(x) ]=[g(x)  d/dx  f(x)- f(x)  d/dx  g(x)]/[g(x)]^2
=   [cos x  d/dx  sin x-sin x  d/dx  cos x]/(cos x )^2
[ d/dx  sin x=cos x    and d/dx  cos x= -sin x ]

=  [cos x .cos x- sin x.  (-sin x)  ]/(cos x )^2

=  [cos ^2  x+  sin ^2 x]/cos^2 x

                = [cos^2 x/cos^2 x +sin^2 x/cos^2 x ]                  [we have,sin^2 x/cos^2 x =tan^2 x  ]

=  1+tan^2 x
           
= sec^2 x
 
       d/dx  tan x= sec ^2 x


Derivative of Tan Inverse:
To find the derivative of an inverse of tangent function, let us take
 y = tan-1x
which would give us,
tan y = tan (tan-1x)
tan y = x            
      Differentiating on both sides with respect to y
d/dy  tan y=dx/dy
    Using the chain rule, we get
d/dy  (tan (y) )  dx/dy=1
1/cos^2 y   dy/dx=1
dy/dx=cos^2 y
y’ = cos2(tan x)                     [y=tan x]
Let us now consider a right triangle,

In the above triangle, tan (y) = x  

 y = arc tan(x)

According to the Pythagorean Theorem, hypotenuse h = v(x^2+1)

Now, we can compute:

cos y=1/v(x^2+1)

Squaring on both sides, we get

cos^2 y=[1/v(x^2+1)]^2

                             cos^2 y=1/(x^2+1)

So, dy/dx=1/(x^2+1)                                [dy/dx=d/dx  tan^(-1)  x,y=tan^(-1) x   ]  

and hence,  d/dx  tan^(-1)  x =1/(x^2+1)

Derivative of Tan x or[d/dx  tan x]:

We know that, tan x = sinx/cosx

Let us derive tan x, with respect to x

Using the Quotient Rule for Derivatives

 d/dx  [f(x)/g(x) ]=[g(x)  d/dx  f(x)- f(x)  d/dx  g(x)]/[g(x)]^2 

Here, f(x) = sin x and g(x) = cos x

=   [cos  x  d/dx  sin x-sin x  d/dx  cos x]/(cos x )^2

[ d/dx  sin  x=cos x    and d/dx  cos x= -sin x ]

 

=  [cos  x .cos x- sin x.  (-sin  x)  ]/(cos x )^2 

=  [cos ^2  x+  sin ^2 x]/cos^2 x

                      = [cos^2 x/cos^2 x +sin^2 x/cos^2 x ]                  [we have,sin^2 x/cos^2 x =tan^2 x  ]

=  1+tan^2 x

             

= sec^2 x   

       d/dx  tan x= sec ^2 x

Derivative of Tan x Inverse [tan-1(x)] or arc tan(x)):

To find the derivative of an inverse of tangent function, let us take

 y = tan-1x

tan y = tan (tan-1x)

tan y = x              

      Differentiating on both sides with respect to y

d/dy  tan y=dx/dy

    Using the chain rule, we get

d/dy  (tan (y) )  dx/dy=1

1/cos^2 y   dy/dx=1

dy/dx=cos^2 y

y’ = cos2(tan x)                     [y=tan x]

consider a right triangle,

 In the above triangle, tan (y) = xy = arc tan(x)According to the Pythagorean Theorem, h = v(x^2+1)Now, we can compute: cos y=1/v(x^2+1)From this, we get cos^2 y=[1/v(x^2+1)]^2                             cos^2 y=1/(x^2+1)     [cos^2 y=dy/dx] dy/dx=1/(x^2+1)                                [dy/dx=tan^(-1) x ]    d/dx tan^(-1)  x =1/(x^2+1)   or  d/dx arc tan(x) = 1/(x^2+1)  

Introduction To Scientific notation

In our real life we come across many situations and numbers which are very large. Like let us take an example , the speed of light in vacuum is  299792.5 km per second. It is approximated as 300000 km per second or as 300,000,000 meter per sec and similarly if we take one more example of  mass of the earth is 5,976,000,000,000,000,000,000,000 kg. These type of numbers are very difficult to read and write. For making them more convenient we write these number by using exponents with base 10 .like speed of light in vaccum we can write as  3 x 10⁸ meter per second. such a  form of a number is known as the standard form.

Scientific Notation

Definition of Standard form 
Mathematically definition of standard form : A number is said to be in standard form  , if it is expressed as the product of a number between 1 and 10 ( including 1 but excluding 10) and a positive integer power of 10.

The standard notation is nothing but the regular form of number whereas the standard form of a number is known as scientific notations

How to do scientific notation
We use the following method to write large number in the scientific notation
• Obtain the number and move the decimal to the left  till we get just one digit left to the decimal point
• Write the given number as the product of the number  so obtained and 10ⁿ , where n is the number of places the decimal point has been moved to the left.
• If the number is between 1 and 10 , then write it as  the product of the number itself and 10⁰.
Let us take some example to see how to do scientific notation

Examples based on scientific notation
Example: For given number , convert to scientific notation
(i) 393568
       To convert it into scientific notation we will use following steps:
     •   Here  we will move the decimal point to five places to obtain a number  where there is just one digit to the left of decimal point
           Therefore  393568= 3.93568  x 10 ⁵
(ii) In  this example  we will learn procedure of adding scientific notation
            3 × 103         and            4.2 × 10⁴
          For addition of the given scientific notation ,
=>  3 x 10³  + 4.2 x 10⁴
=> Step1: we will change it to same indicies
=> 0.3 x 10⁴ + 4.2 x 10⁴
=> (0.3 + 4.2)x 10⁴
=> 4.5 x 10⁴ ans