MathLabs

Problem 2

Find all functions f:R→Rf:\mathbb R\to\mathbb R such that f(x2+f(y))=y+f(x)2f(x^2+f(y))=y+f(x)^2 for all real x,yx,y.
Step 1 of 5: Record the substitutions at zero
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In plain words

Record the substitutions at zero

t=f(0),f(t)=t2,f(x2+t)=f(x)2,f(f(y))=y+t2t=f(0),\quad f(t)=t^2,\quad f(x^2+t)=f(x)^2,\quad f(f(y))=y+t^2
Detailed analysis

Write t=f(0)t=f(0). Substituting x=y=0x=y=0, then y=0y=0, and then x=0x=0 in the given equation gives respectively f(t)=t2f(t)=t^2, f(x2+t)=f(x)2f(x^2+t)=f(x)^2, and f(f(y))=y+t2f(f(y))=y+t^2 .