Problem 5

Let

\mathbb{R}

be the set of real numbers. Determine all functions

f:\mathbb{R}\to\mathbb{R}

satisfying the equation

f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)

for all real numbers

x

and

y

Step 1 of 5: Every

y=1

substitution produces a fixed point

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In plain words

Setting $y=1$ collapses the product term $xy$ into just $x$ , revealing a whole family of fixed points.

P(x,1):\ f(x+f(x+1))=x+f(x+1)

Detailed analysis

Let $P(x,y)$ denote the assertion $f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)$ . Taking $y=1$ : $P(x,1)$ gives $f(x+f(x+1))+f(x)=x+f(x+1)+f(x)$ , so $f\big(x+f(x+1)\big)=x+f(x+1)$ . That is, the number $x+f(x+1)$ is a fixed point of $f$ , for every real $x$ .