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.