Question

quick question :
if we are given f(f(x)) = f(x), f:X->X
is it okay to conclude f(x) = x ?

Answer 1

f(x)=C or f(x)=0 seem to be possibilities as well.

Answer 2

Or actually the math floor, math ceiling, and absolute value of x functions also have this property.

Answer 3

http://en.wikipedia.org/wiki/Idempotence

Answer 4

Yes you can

Answer 5

If I substitute f(x) = y then :

\[\large{f(y) = y}\]

Which is similar to

\[\large{f(x) = x}\]

Answer 6

Provided that domain and range are same ;) :P

Answer 7

f(x) = c
f(f(x)) = f(c) = c

oh my bad yes :o

whats wrong with below argumennt :

\[f(f(x)) = f(x)\]
taking inverse both sides gives
\[f(x) = x\]

right ?

Answer 8

yeah

Answer 9

Well both are correct but ensure that domain and range satisfy the conditions

Answer 10

That's a cool way of doing it, I was wondering how you came to that answer. Nice. =)

Answer 11

\(\large{fof(x)}\) and other composite functions are very stubborn while dealt with domain and range

Answer 12

@ganeshie8 Solving the combinatorics question ? ;)

Answer 13

yes trying lol
so f(x)=x doesn't cover all the functions for which f(f(x)) = x is it ?
cuz f(x) =c is certainly an outlier

Answer 14

Yes inverse of function will exist only if certain domain and range conditions are fulfilled.

It would be best if you use your question along with the restrictions assumed to be true

Answer 15

Finally you can find out the total number of possibilities if you need :)

Answer 16

Oh yes inverse need not exist for f, thats another part ive missed
this info is given in the question : `f : X->X, X has "n" elements`

Answer 17

that wiki link has the formula for total # of functions possible :

\[\large \sum \limits_{k=0}^n\binom{n}{k}k^{n-k}\]

Answer 18

Yup correct formula :) But would it help ?

Answer 19

If that info is given then inverse will exist

Answer 20

\[\large \sum \limits_{k=0}^n\binom{n}{k} = 2^n\] refers to powerset, and i think it represents the total number of functions for which f(x) = x

Answer 21

* refers to powerset, and i think it represents the total number of functions of form : f(x) = x

Answer 22

\(\large k^{n-k}\) is somehow accounting for other functions :o

Answer 23

One more thing: for inverse to exist the function must be one-one and onto. Ensure that this condition is also fulfilled

Answer 24

Or you can just use a formula to find out the number of such functions :)

Answer 25

Then state that out of these possible functions one is f(x)

Not sure if it is going to help... I am not that good at combinatorics :D

Answer 26

so if i understand it correctly :

f(x) = x essentially represents all the invertible funcitons for which f(f(x)) = f(x)
is true
and there could be other functions without inverses of form f(f(x)) = f(x) ?

Answer 27

I am not sure... the problem is if I think that fof(x) = f(x) is true then it appears to me that even stating that f(x) belongs to the domain of x, then it should be invertible because in any other case if would not be true *shrug*

Answer 28

Because the equation probably automatically ensures that the function is invertible. If we think along this path then it should imply that function is both one one and onto. While this can provide some more results regarding domain and range of the function, it probably would not be able to solve the question

Answer 29

I see... in that case we can safely take inverse both sides and conclude f(x)=x right ?

Answer 30

Yeah I think so

Answer 31

then that formula in wiki must be representing all the funcitons of form : f(x) = x

Answer 32

Probably

Answer 33

Oh wait, what about the outlier : f(x) = c
f(x) = x doesn't capture this function

Answer 34

This is the main problem that I too am facing

Answer 35

then good :) we're one same page xD

Answer 36

:D

Answer 37

I was on the same page from yesterday evening IST

Answer 38

:P

Answer 39

haha same here lol
that wiki link looks useful, i am going thru idempotence stuff

Answer 40

Okay :)

Hope that you find something helpful from the link