Problem 2

Find all functions

f:\mathbb R\to\mathbb R

such that

f(x^2+f(y))=y+f(x)^2

for all real

x,y

Step 5 of 5: Use nonnegativity to identify the function

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In plain words

Use nonnegativity to identify the function

f(z)=-z<0\quad\text{and}\quad f(z)=f(w^2)=f(w)^2\ge0

Detailed analysis

Suppose $f(x)=y\ne x$ . If $y>x$ , set $z=y-x>0$ ; if $y<x$ , set $z=x-y>0$ . Additivity gives $f(z)=-z<0$ in either case. Choose $w$ with $w^2=z$ . But $f(z)=f(w^2)=f(w)^2\ge0$ , a contradiction. Therefore $f(x)=x$ for every real $x$ , and direct substitution verifies it.