Inverse Function

If a set  obtained  by  interchanging  the  co-ordinates   of 

each ordered pair  of  a function  is  a  function  , then  the given  function  is  an  invertible  function  and  the   new  function  is  the  inverse  function  .

A  function  has  inverse  if and only if it  is  a  one-to-one  

function .

 E.g.

*

{(1,2),(3,4),(5,1)} → {(2,1),(4,3),(1,5)}

It is  a  invertible  function and its inverse  function  

{(2,1),(4,3),(1,5)}.

*

{(1,2),(3,2),(5,1)} →  {(2,1),(2,3),(1,5)}

It is a function but not invertible  function  . So  Its can not  do  inverse .

Definition:  Lets f  be  a  One-to-One  function with  domain X and  range  Y .The  inverse  of  f  is  the  function  g  with  domain  Y and  Range X  for which

 

        f(g(x)) = x  for every x in Y   

  and  g(f(x)) = x for every x in X

So f  is not  One-to-One ,then it has no inverse function.

Procedure of finding  f^-1(x)

    1) Check whether  the  function  is  One-to-One

    2) Write the function y=f(x)

    3) Solve for x (in terms of y)

    4) Interchange x and y

    5) Write y= f^-1(x)

Properties :

i)                 Domain of f = Range of f^-1

ii)              Range of f = Domain of f^-1

iii)           f^-1  is One-to-One.

iv)           Inverse of f^-1 is  f

Example :

y = f(x) =(x+1) / (2x-5)

Domain of f = R – {5/2 }

Here, y = (x+1)/(2x-5)

      => 2xy-5y=x+1

      => 2xy-x=5y+1

      => x(2y-1)=5y+1

      So x = (5y+1) / (2y-1)

Interchanging x and y , we get , 
       y = (5x+1) / (2x-1) = f^-1(x)

Domain of f = R – {1/2 }

Range of f = R – {5/2}

Graphs of f and f^-1

The   graph  of  f and f^-1 are reflections  of  each other  in the  line  y = x . Tn the  other  words , the graphs  of  f and  f^-1  are  symmetric  with  respect  to  the  line  y=x .

Such  as ,  y = x³

One-to-One function

A function  f  is  said  to  be  One-to-One  function  if  each  number  in  the  range  of   f   is associated with exactly  one  number  in   its  domain  X.

Another ,  A  function  f   is  called  One –to-One  function  if  it  never  takes on the  same value  twice. .

That’s ,

                  f(x₁) ≠ f(x₂)    ; whenever  x₁≠x₂

One-to-One  function can written like 1-1 .It is One-to-One  function if it passes both the vertical line test and the horizontal line test.Another way of testing whether a function is 1-1 is given below,

        Test  for   One-to-One   function  :

                    If  f(a) = f(b) ,implies that  a =  b , then  f  is  One-to-One . 

                Suppose we can  try  to prove  g(x) = 3x - 2 is One-to-One .

                              see if  g(a) = g(b) , implies that  a = b,

                                      3a-2 = 3b -2

                                          3a = 3b

                                            a = b

                                       Thus  g  is  1-1 .

Example1:

Is the function   f(x) = x³  one-to-one?

Solution:  If   x₁ ≠ x₂   ,then   x₁³ ≠ x₂³   (two different numbers can’t have  the  same  cube).

Therefore , f(x)=x³ is one to one function..

Example 2: 

Is the function f(x)=x²one-to-one?

Solution : This  function  is  not  One-to-One  because  ,  

              G(1)=1=G(-1)

 And   so  1  and   -1   have  the  same  output.

Example 3:

Is the function s(x)= Sin(x)  one-to-one?

Here         Sin(0) = 0

                Sin (2π)=0

                Sin(4π) =0

So  s(x) = Sin(x) is not One-to-One function.

*But if there give a restriction in value of x , such thet (0 ≤ x ≤  π/2).

Then   s(x) = Sin(x) is One-to-One .

Example 4:

If A={(3,1),(2,1)}

here D={3,2}

       R={1}

range same for every domain .

so its not One-to-One.

Horizontal line test

A Horizontal line (y=c) can intersect the graph of f , a one-to-one function in at most one point .If intersects at more than one  point then it is not , one to one function .

So,   A function is one-to-one if and only if no horizontal line intersects its graph more than once.  Its called Horizontal line test.

Example 1: Is the function f(x)=x³   one-to-one?

Solution: Form the figure 3 we can see no horizontal line intersect the cube more than once.

Therefore , by Horizontal line test , f is one-to-one function  

Example 2: Is the function f(x)=x²  one-to-one?

Solution: Form the figure we can see  horizontal line intersect the cube more than once.

Therefore  , by Horizontal line test , f isn't one-to-one function  

Even function And Odd function

Even function :

If f(-x)= f(x) ,then f is a even function ,which is related to symmetry about y-axis,if is replaced by -x and produces an equivalent equation.
For instance, the function f(x)=x^2 is even because

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

The geometric significance of an even function is that its graph is symmetric with respect to the y axis.This means that if we have plotted the graph of for x>=0 ,we obtain the entire graph simply by reflecting this portion about the
 y-axis.  Figure 19 shows..

Odd function :

If f(-x)= -f(x) ,then f is a odd function ,which is related to symmetry about the origin,if and only if replacing both x by -x and y by -y in its equation produces an equivalent equation.
For instance, the function f(x)=x^3 is odd because

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

The geometric significance of an odd function is that its graph is symmetric with respect to the origin.This means that if we have plotted the graph of for x>=0 ,

we obtain the entire graph simply by reflecting this portion about the origin.
Figure 20 shows..

Symmetry

A plane curve is Symmetric about,

i)about y-axis,if is replaced by -x and produces an equivalent equation.

 such as  given a equation  y=x^2
                                       or y=(-x)^2
                                       so y=x^2

                     so its symmetric about y-axis.


ii)about x-axis,if is replaced by -y and produces an equivalent equation.

such as  given a equation x=y^2
                                     or x=(-y)^2
                                     so x=y^2
 
                  so its symmetric about x-axis.


iii)about the origin,if and only if replacing both x by -x and y by -y in its equation produces an equivalent equation

such as  given a equation  y=x^3
                                    or -y=(-x)^3
                                    or -y=-x^3
                                    so  y=x^3
   
                 so its symmetric about origin.

Translations , Reflections ,Stretches And Compression

Translations of functions

Suppose that y = f(x)  is a function and c is a positive constant.
Then the graph of,

a) y = f(x)+c is the graph of f shifted vertically Up  c units.
b) y = f(x)-c is the graph of f shifted vertically Down  c units.
c) y = f(x+c) is the graph of f shifted horizontally to the left  c units.
d) y = f(x-c) is the graph of f shifted horizontally to the right  c units.

Show figure 1 :





Reflections of function:

Suppose that y = f(x)  is a function. Then the graph of ,

1) y = -f(x) is the graph of f reflected in the x=axis.
2) y = f(-x) is the graph of f reflected in the y=axis.

Show figure 2

Stretches And Compression : 

Suppose that y = f(x)  is a function and c is a positive constant.
Then the graph of,

1) y =c f(x) is the graph of f
            i) vertically stretched by a factor of c if c > 1
           ii) vertically compressed by a factor of (1/c) if  0 < c < 1.
2) y = f(cx) is the graph of f
            i) horizontally  stretched by a factor of (1/c) if  0 < c < 1.
           ii) horizontally compressed by a factor of c if c > 1.

Show figure 2 

Power Function

A function of the from f(x)=x^n is called a power function.
There n is a rational real number .
 The domain of power functions depend on the value of n.
 Such as
 f(x)=x^n

         f(x)=x^2
             Domain=R

         f(x)=x^(1/2)=Sqrt[x]
             Domain=[0,+infinity)

         f(x)=x^(-1)
             Domain =R-{0}

When n is a positive integer
The graphs of f(x)=x^n for n=1,2,3,4  and 5 are shown below. (These are polynomials with only one term.)

The general shape of the graph of f(x)=x^n depends on whether n is even or odd.If n is even, then f(x)=x^n is an even function and its graph is similar to the parabola f((x)=x^2. If n is odd, then f(x)=x^n is an odd function and its graph is similar to that of f((x)=x^3.


when 1/n , where n is positive integer
  its   called Root function.

when n = -1
its called reciprocal function.

Composition of function:

Given functiond f and g ,the composition of f and g denoted by 'fog' is the function defined by -
        (fog)(x)=f(g(x))         ;     (gof)(x)=g(f(x))

The domain of fog is defined to consist of all x and domain of g for which g(x) is the domain of f(x).

Example::

 here, f(x)=x^2+3
          g(x)=Sqrt[x]

 Now, (fog)(x)=f(g(x))
                       =f(Sqrt[x])
                       =x+3

 domain of g=[0,+Infinity)
 domain of fog=[0,+Infinity)


Again , (gof)(x)=g(f(x))
                         =g(x^2+3)
                         =Sqrt(x^2+3)
             
  domain of , f=(-Infinity,+Infinity)
  domain of ,gof=(-Infinity,+Infinity)

Vertical Line Test

             A curve in the XY plane is the group of some function f if and only if no vertical line intersects the curve more than once.

For example, the parabola  x=y^2-2 shown in Figure (a) is not the graph of a function x of because, as you can see, there are vertical lines that intersect the parabola twice. The parabola, however, does contain the graphs of two functions of x . Notice that the equation x=y^2-2 implies y^2=x+2 , so
y=Sqrt(x+2) and -[Sqrt(x+2)] .
Thus the upper and lower halves of the parabola are the graphs of the functions f(x)=Sqrt(x+2) and g(x)= -[Sqrt(x+2)]  [See Figures (b) and (c).] We observe that if we reverse the roles of x and y , then the equation
x = h(y) =y^2-2 does define x as a function of y (with y as the independent variable and  x as the dependent variable) and the parabola now appears as the graph of the function h .

Function

  A Function is a rule that takes certain numbers as inputs and 

assigns to each a definite output numbers.

    Domain : 

  The set of all input numbers are called the ' Domain' of the 

function.

    Range :

   The set of resulting output numbers are called the ' Range' of 

the function.

       

       Input is called the independent variable and Output is 

called  the dependent variable. 

       

       Example :

        y = f (x) = x^2 + 2

Here , y = output = dependent variable

               x = input = independent variable

     Domain = possible set of x

     Range = Possible set of y

In this function 

Domain = 0, -1, 1, 2, ... ... ... ..

Range = 2, 3, 3, 6 ... ...

Understanding the most beautiful equation in Mathematics

 Euler was one of the most influential and prolific mathematicians in history. He had published over 800 papers and 20 books, making him the greatest contributor in mathematics. Referred as the Mozart of Mathematics, Euler left hardly any area of Mathematics untouched, contributing to various field like mathematical analysis, number theory, mechanics and hydrodynamics, cartography, fluid dynamics and topology. In this article, we'll try to understand the most beautiful equation in all of mathematics:

eiπ+1=0

It connects the five most important constants of mathematics and three most important mathematical operations - addition, multiplication and exponentiation. So, how did Euler arrived at this result?

The Euler's constant e is defined as lim(1+1n)nas n approaches infinity.It's approximate value is equal to 2.71828. In his most influential work, Introductio in analysin infinitorum, Euler defined the function e^x in analysis as: ex=lim(1+xn)n as n tends to infinity. So, we get: ex=1+x1!+x22!+x33!+...

This is the known series for ex$e^x$ Euler's brilliant mathematical mind replaced the real variable x with ix were i = −1−−−√$\sqrt{-1}$ .

So, we get: eix=1+ix1!+(ix)22!+(ix)33!+..

We know that square of i is equal to -1. So, replacing subsequent values for i3,i4,i5...$i^3,i^4,i^5...$ , we get:

eix=1+ix1!−x22!−ix33!+.

On separating real and imaginary parts, we have: eix=(1−x22!+x44!+...)+i(x−x33!+x55!+...)${\mathrm{<br>}}^{}$

So, we got two trigonometric series of cosx and sinx respectively. Hence, eix=cosx+isinx

If we put, x=π$x=\pi$ , we get:

eiπ=−1ascosπ=−1

and sinπ=0

or

eiπ+1=0

So, this was the story of the creation of most beautiful equation in mathematics. I would like to end this article by a quote from professor Keith Devlin - "Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's Equation reaches down into the very depths of existence."