MathLabs

Problem 5

Let R\mathbb R denote the set of all real numbers. Find all functions f:R→Rf:\mathbb R\to\mathbb R such that (i) only finitely many s∈Rs\in\mathbb R satisfy f(s)=0f(s)=0, and (ii) f(x4+y)=x3f(x)+f(f(y))f(x^4+y)=x^3f(x)+f(f(y)) for all x,y∈Rx,y\in\mathbb R.
Step 1 of 3: Extract the basic identities
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P(x,y):f(x4+y)=x3f(x)+f(f(y))P(x,y): f(x^4+y)=x^3f(x)+f(f(y))
Detailed analysis

From P(1,0)P(1,0), f(f(0))=0f(f(0))=0. From P(0,y)P(0,y), f(y)=f(f(y))f(y)=f(f(y)); taking y=0y=0 gives f(0)=0f(0)=0. Thus f(f(y))=f(y)f(f(y))=f(y) for all yy.