Question

Find all$$f:R→R$$such that$$ f(x^2+f(y))=(f(x))^2+y,∀ x,y∈R$$
Help me to understand this type of questions.I am new to this pattern.

Thanks in advance!

Answer 1

@rational

Answer 2

@Michele_Laino @rational please help :)

Answer 3

is the answer (x+4)/3?

Answer 4

hello..

Answer 5

answer is$$ f(x)=x$$

Answer 6

ok..

Answer 7

@Michele_Laino hey will you please help

Answer 8

here I'm

Answer 9

yes
will you please wait for sometime Michele please :)

Answer 10

ok!

Answer 11

okay I am here :) Thanks sister :) @rvc

Answer 12

Thanks @Michele_Laino for waiting so patiently :)

Answer 13

@Michele_Laino will the function be an indentity function.

Answer 14

I think that your equation can be viewed as an operational equation

Answer 15

namely a functional equation

Answer 16

yes it is from the olympiad prep materials. And a question from the functional equation section.

Answer 17

there exist an operator, say T, such that, your equation can be written as below:
\[T\left( f \right) = ...\]
so, we have to know who is our operator T

Answer 18

taking x=0 we have $$\begin{align} \forall x,y\in \Bbb R^2&& f\left(f\left(y\right)\right)=y+f(0)^2=y\tag2\end{align}$$

which means that f is one one and onto. can it be ???

Answer 19

I'm not sure

Answer 20

for y=o we will have$$ f(x^2)=(f(x))^2$$

Answer 21

why f(0)=0?

Answer 22

please wait a moment, someone is calling me to my phone

Answer 23

here I'm

Answer 24

using the Taylor expansion, at the left side, we can write:
\[\Large f\left( {{x^2} + f\left( y \right)} \right) = f\left( {{x^2}} \right) + f\left( y \right)\frac{{df}}{{dx}}\]

Answer 25

sorry, that formula is wrong

Answer 26

I guess so ! I am confused with the step.

Answer 27

for example one solution of your problem is the identity function, as you stated before.