# F^2(x)

Solve the equation f(f(x))=f^2(x), where f(x)=x+2, f being a real function.

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We notice that f(f(x)) is the composition law: f composed with f.

f(f(x))= f(x)+2= x+2+2=x+4

f(f(x))= x+4

f^2(x)= (x+2)^2= x^2+2x+4

x^2+4x+4=x+4

x^2+4x+4-x-4=0

x^2+3x=0

x(x+3)=0

If a product between 2 factors is zero, that means that one of them is zero.

**x=0**

x+1=0

**x=-3**

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

Therefore f(f(x+2) = f (x+2) = (x+2)+2 =x+4