For IMO

$\large{ f(xf(x + y)) = f(yf(x)) + x^2}$

Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all real numbers $x,y$ and the equation above.

#Algebra #Functions #Reshare #Sharky #Lakshya

Easy Math Editor

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Put $y=0$ in the given functional equation.

we get $\boxed{f(xf(x))=f(0)+x^{2}}...........(1)$ Now put $x=-x$ in the given functional equation .

We get $\boxed{f(-xf(-x)=f(0)+x^{2}}...........(2)$

Equating $(1),(2)$ we get

$f(-xf(-x))=f(xf(x))$ $-xf(-x)=xf(x)$ $\boxed{f(x)+f(-x)=0}..........(3)$

Now by putting $x=0$ we get $\boxed{f(0)=0}........(4)$ . Now equations $(1),(2)$ transform into

$\boxed{f(xf(x))=x^{2}}...........(I)$

$\boxed{f(xf(x))=x^{2}}...........(II)$

Our main functional equation is $f(xf(x+y))=f(yf(x))+x^{2}$

This implies that degree of $f(yf(x)),f(xf(x+y))$ is $2$ .

This implies that degree of $f(x),f(x+y)=1$ .

Therefore we can write $f(x)=ax+b$ where $a,b$ are real numbers.

But $f(0)=0$ therefore $b=0$ . therefore $f(x)=ax$ . But for $f(x)=ax$ to satisfy $(I)$ $a=1$ is a must condition .

Therefore our final function reduces to $f(x)=x$ .

Therefore the required function is $\boxed{f(x)=x}$ .

Shivam Jadhav - 5 years, 5 months ago

nice but typo.. if you put y=0 in the first line you would get $f(xf(x))=f(0)+x^2$

Aareyan Manzoor - 5 years, 5 months ago

Yeah just saw that now

Department 8 - 5 years, 5 months ago

Nice observations but it did not affect the solution greatly.

Shivam Jadhav - 5 years, 5 months ago

How can u say that in the line above (3) ? I dont think that it is obvious that the given functional equation is injective. If yes then please prove it Also I dont see how do u get f(0)=0. Also arguments like degree of so and so .... are not allowed. What is the function is not a polynomial ? If it is something like exponential then argument fails. Please check the solution again.

Shrihari B - 5 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$