Find all real functions f from #RR-RR# satisfying the relation #f(x²+yf(x))=xf(x+y)# ?

Question

Find all real functions f from #RR->RR# satisfying the relation #f(x²+yf(x))=xf(x+y)# ?

Abigail Allen · Accepted Answer

#f(x)=x#

Making #x=y=0# we have #f(0)=0#

Making #y=0# we have

#f(x^2)=xf(x)# but #f((-x)^2)=f(x^2)=-xf(-x)#

so #f(x)# is an odd function.

Making now #x = -y# we have

#f(x^2-xf(x))=x f(0) = 0#

thus presuming that

#f(x)=0 => x=0#

(remember that #f(x)# is an odd function) we have

#x^2-xf(x)=x(x-f(x))=0#

considering #x ne 0# we have

#f(x)=x#

Verifying

#(x^2+y x)=x(x+y)#

Note. Another way of proof for #f(x)=x# is by doing for #x ge 0#

#f(x) = x^(1/2)f(x^(1/2))= cdots = x^(1/2+1/4+cdots + 1/(2^n))f(x^(1/(2^n)))#

but #lim_(n->oo) x^(1/2^n) = 1# and #lim_(n->oo)(1/2+1/4+cdots+1/(2^n))=1# so

#f(x) = x f(1)# now substituting this result into #f(x^2)=xf(x)# we obtain

#f(x^2)=x^2f(1)->f(1)=1#

Jace Anderson · Answer

undefined The only solutions to the functional equation $ f(x^2 + yf(x)) = xf(x + y) $ are $ f(x) = 0 $ and $ f(x) = x $.

Savannah Anderson · Answer

undefined To find all real functions $ f $ from $ \mathbb{R} $ to $ \mathbb{R} $ satisfying the relation $ f(x^2+yf(x))=xf(x+y) $, we can proceed by substitution and manipulation.

Let $ x = y = 0 $:
$$ f(0) = 0 $$

Let $ y = 0 $:
$$ f(x^2) = xf(x) $$

Now, let's substitute $ x = -1 $ in the equation above:
$$ f(1) = -f(-1) $$

Now, we'll use the equation $ f(x^2) = xf(x) $ and substitute $ x^2 $ with $ x $ in the original equation:
$$ f(x+f(x)) = xf(x+1) $$

Now, let's substitute $ x = 1 $ in the above equation:
$$ f(1+f(1)) = f(2) $$

And from the previous deduction, we know $ f(1) = -f(-1) $, so:
$$ f(-1+f(-1)) = f(0) = 0 $$

This implies $ f(-1) = -1 $.

Now, let's set $ x = 1 $ and $ y = -1 $ in the original equation:
$$ f(1+(-1)f(1)) = f(0) = 0 $$

So, $ f(0) = 0 $.

Now, using $ f(x^2) = xf(x) $ and $ f(1) = -1 $, we can derive that $ f(x) = x $ for all $ x $ in $ \mathbb{R} $.

Find all real functions f from #RR-&gt;RR# satisfying the relation #f(x²+yf(x))=xf(x+y)# ?

Find all real functions f from #RR->RR# satisfying the relation #f(x²+yf(x))=xf(x+y)# ?