Question

Does the inverse function of the following real valued function of real variable exist?Give reason.
f(x)=[x]

Answer 1

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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कारण  f(x)=|x|

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Answer 2

HELLO DEAR,

GIVEN:- Does the inverse function of the following real valued function of real variable exist?Give reason.

f(x)=[x]

SOLUTION:- we known that inverse function f(x) = y if f^-1(y) = x.

So, f(x) = [x]

Where [x] is a greater integer function.

And the property of greater integer function is,

f(x) = [x]

[1] = 1

[1.2] = 1

[1.3] = 1

[1.4] = 1

[1.5] = 1

But if we f^-1(x) = [x]

[1] = 1

[1] =/= 1.2

So, inverse of greater integer function is many value function .

Therefore ,it does not exist.

I HOPE IT'S HELP YOU DEAR,

THANKS.