MathLabs

Problem 5

Find all functions f:R+→R+f:\mathbb{R}^+\to\mathbb{R}^+ such that (z+1)f(x+y)=f(xf(z)+y)+f(yf(z)+x)(z+1)f(x+y)=f(xf(z)+y)+f(yf(z)+x) for all positive real numbers x,y,zx,y,z.
Step 6 of 8: ff is an involution
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f(f(y))=yf(f(y))=y
Detailed analysis

Setting x=1x=1 in yf(x)=f(xf(y))yf(x)=f(xf(y)) and using f(1)=1f(1)=1 gives y=f(f(y))y=f(f(y)) for every y>0y>0, so ff is an involution.