Question

Does the inverse function of the following real valued function of real variable
exist ? Give reasons.
f(x)=(x) ​

Answer 1

Given:

Real valued function of real variable $f(x)=x$.

To Find:

Does the inverse function exist?

Solution:

for given function ;    $f(x)=x$

check for one-one;

let,

$f(x)=f(y)\\x=y$          for real variable x and y.

therefore, f is one-one.

Check for onto;

let, for any real variable 'y' ,

$y=f(x)\\x=f^{-1}(y)$            for real variable x.

therefore, f is onto.

⇒ f(x) is invertible.

⇒ inverse of given function exist.

Hence, inverse function of given real valued function of real variable exist.  

Answer 2

Given : f(x)=[x]

To Find : Does the inverse function  of real variable exist

Solution:

f(x)=[x]

[x]   = Greatest integer function

x = 1  => f(1)  = 1

x= 1.01  => f(1.01) = 1

x = 1.2 => f(1.2) = 1

x = 1.99 => f(1.99) = 1

f(1) =  f(1.2) but 1#1.2  Hence function is not one to one

As function is not one to one  so function  is not bijective  

A function is said to be invertible iff it is bijective

Since  function   is not bijective  

Hence the inverse does not exists

=>  the inverse function of the f(x)=[x] of real variable does not exist

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क्या निम्नलिखित वास्तविक चर के वास्तविक-मानित फलन का प्रतिलोम फलन प्राप्त होग

कारण  f(x)=|x|

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