Inverse The Floor

Algebra Level 2

$\large f(x)=\lfloor x \rfloor$

Let the function $f$ be defined for all real numbers $x$ .
Is it possible to define the inverse function $f^{-1}(x)$ ?

Notation : $\lfloor \cdot \rfloor$ denotes the floor function .

Yes No

2 solutions

Hung Woei Neoh
Jul 10, 2016

Relevant wiki: Inverse Functions

Functions can only have one image for each object, that is, it can have a

one-to-one relation. Eg. $f(x)=x$
many-to-one relation. Eg. $f(x)=x^2$

For a function to have an inverse function, the function must be a one-to-one relation. This is because when many-to-one relations are inversed, they become one-to-many relations, which are not functions.

Note that $f(x)=\lfloor x\rfloor$ is a many-to-one relation:

$f(1)=f(1.2)=f(1.5)=1$

This implies that this function does not have an inverse function $f^{-1}(x)$ . The answer is $\boxed{\text{No}}$

Domain must be specified first of all. Example sine is also not invertible, we restrict the domain so that it is invertible .And in mathematics there is nothing pointless as you said,because it is asked that is it possible to define f such that it's inverse exists. So yes it is possible. Possibility and common sense are different things.

Ravi Dwivedi - 4 years, 11 months ago

Well, this I agree. Although it's very stupid and pointless (mathematicians do lots of pointless thinking) to do so, we actually CAN define it in such a manner. So yes, I'm with you on this one. Note that you can file a report to notify the problem owner about it

Hung Woei Neoh - 4 years, 11 months ago

It can have inverse if I define it from set of natural numbers to set of natural numbers f:N->N

Ravi Dwivedi - 4 years, 11 months ago

If the domain is integers, then yes, this function has an inverse. Nothing is said here, so we assume that the domain is all real numbers. Besides, it's totally pointless to define a floor function for integers only

Hung Woei Neoh - 4 years, 11 months ago

It may be stupid to the common sense but stupid itself is a vague term and in mathematics, there is no place for these terms as far as defining functions is concerned and about giving examples. A function is something which satisfies it's definition. It is asking about possibility so we cannot assume what's the domain(such as the set of all real numbers)

Asking for possibility gives the question the point I m saying.

Ravi Dwivedi - 4 years, 11 months ago

Thanks for your valuable comments.I am going to report it

Ravi Dwivedi - 4 years, 11 months ago

Kushal Bose
Jul 10, 2016

$f(x)=\lfloor x \rfloor$ is not a bijective mapping

$\lfloor 3.4 \rfloor =3$ and $\lfloor 3.7 \rfloor=3$

So, it has multiple pre-images.

No inverse exists.

Inverse The Floor

2 solutions

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