Tripling an involution (Functional equation)

Question

Here's a standard trick which is claimed by Evan Chen in one of his handouts: Introduction to Functional Equations:

Tripling an Involution: If you know something about $f(f(x))$, try applying it $f(f(f(x)))$ in different ways. For example, if we know that $f(f(x))=x+2$, then we obtain $$\boxed{f^3(x)=f(x+2)=f(x)+2}$$

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Now, I fail to understand how $f^3(x)\equiv f(x)+2$. May I get an explanation of how that transformation just took place.

Answer 1

$$f^3(x)=f(f(f(x)))=f(f(y))=y+2=f(x)+2$$ where y=f(x)