Solving a Quick and Easy Homemade Functional Equation

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[randomjin]

Set z = x/f(y) then x = zf(y) and we have f(z) = zf(y)/y. For y = 1 we have f(z) = zf(1). Since f(1) is a constant it means f(x) = Cx

[Jha-s-kitchen]

Yes! Please show us how do you come with these homemade problems

[garvkumar]

Replace x by f(y)/x to form the eqn.
f(1/x)=f(y)/xy
replace 1/x by x then
f(x)=(f(y)/y)x
here f(y)/y is just constant therefore
f(x)=kx ( where k=f(y)/y) ).

[crazycat]

Wait! Why can't x/f(x) be a variable, but only a few numbers, for example x/f(x)=3 or 4?

[Your_choise]

The way I solved it:

F(1/f(1))=1 for convenience let 1/f(1)=k-> f(k)=1
f(x)=f(x/1)=f(x/f(k))=x/k
so f(x)= cx for some c because k is a constant
If c is not 0 then f(x/f(y))=cx/f(y)=cx/cy=x/y so all non zero c works
Thus f(x)=cx for some non-zero c

[stmmniko]

set x = f(y) and you get instant solution

[alfiehellings]

Your method is incorrect. At 2:46 you assumed that f is an injective function which may not be valid. The conclusion you draw turns out to be correct (as I will discuss below) but the point is that the reasoning you used to reach it is invalid.

Correct method:


Choose y such that f(y) is non-zero. This is possible since if f is identically 0 then the formula does not really make sense. Set x=0 to obtain f(0)=0. Keep the same choice of y and suppose f(z)=0=0/y. Then z=0/f(y)=0, which is enough to prove that f(x)=0 if and only if x=0.

Now, again choose y such that f(y) is non-zero, and set x=f(y). We obtain that f(1)=f(y)/y, so f(y)=y*f(1), provided f(y) is non-zero. Using the previous paragraph, if f(y)=0 then y=0, and since f(0)=0=0*f(1), it turns out that this formula is valid for any real value of y. ie f(y)=y*f(1) for all y\in R.

It is now simple to check that any function of this form satisfies the given equation.

[Errenium]

my first instinct is to try *x=f(1)u* and *y=1*.
this gives f(u) = f(1)u.
applying this knowledge to the functional equation while leaving x and y free demonstrates that one may freely choose f(1) to be any nonzero real number, so f(x) = kx

[MrLidless]

Easy to see f is continuous. So set not x, not y, but f(y)=1 and you get right there.

[thebatmanmagicianparapente]

I think it is badly defined, what is x what is y ?

[mcwulf]

Put x = f(y) and you get f(y) = f(1)y.
So f(x) = kx. Where k is anything.

[miguelmartincorreacoronel]

Hola amigos.
Les dejo este enlace.
Saludos.

[Dan-cwxu]

What? You made this up and no golden ratio?