Finding functions involving f(x)

[___kshro10]

The problem is

Find all functions f:R(all real numbers) —> R such that

f(f(x)) + xf(x) = 1

My initial thought was plugging in values such as x=0, x=1, and see if there was any pattern or if it lead me to any equations that I could solve.
The answer is that there are no such functions, but I just can’t wrap my head around how to explain it.
I am just completely stuck on this problem, and I was wondering if anyone can give me advice on how to get started, or explain why there are no solutions.
Thank you in advance.
(If this post is in a different category, please let me know, I will take it down and post it under the right one.)

 

[lev888]

The problem is

Find all functions f:R(all real numbers) —> R such that

f(f(x)) + xf(x) = 1

My initial thought was plugging in values such as x=0, x=1, and see if there was any pattern or if it lead me to any equations that I could solve.
The answer is that there are no such functions, but I just can’t wrap my head around how to explain it.
I am just completely stuck on this problem, and I was wondering if anyone can give me advice on how to get started, or explain why there are no solutions.
Thank you in advance.
(If this post is in a different category, please let me know, I will take it down and post it under the right one.)

How did plugging in values lead you to your conclusion?

 

[___kshro10]

How did plugging in values lead you to your conclusion?

My teacher just told the class that the answer was no such functions, but they did not explain how to get to the answer. Plugging in values did not help as much as I thought, because it just puts out f(f(n)) + nf(n) =1.

Sorry for the confusion.

 

[lev888]

My teacher just told the class that the answer was no such functions, but they did not explain how to get to the answer. Plugging in values did not help as much as I thought, because it just puts out f(f(n)) + nf(n) =1.

Sorry for the confusion.

Plugging in values may tell you something about the function.
It's not obvious how to derive this function, but you can figure out some of its characteristics. E.g. you can prove that f(x) = 0 has no solutions (see a similar problem here: https://math.stackexchange.com/questions/408132/find-functions-f-such-that-ffx-xfx1).

To prove that no such functions exist you may be able to use proof by contradiction. E.g. assume such function exists. Then, suppose you could prove that
1. as x -> +infinity the limit is +infinity
2. as x -> -infinity the limit is -infinity
But this would contradict that there is no x-intercept (since the function is defined for all reals).
The above is just an illustration, not an actual proof for this problem.

 

[Steven G]

If the left hand side equals 1, then the derivative of the left hand must be ....? Will this help?