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This question seems a bit stupid but I am really confused: Let $f$ be a function and $g(z):=f(z)-f(2z)$.

What is $g(-1/z)$? Is it $f(-1/z)-f(-2/z)$ or $f(z)-f(-1/(2z))$?

Thank you!

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    $\begingroup$ You might find it helpful to involve more symbols. Define $g(x)=f(x)-f(2x)$. Let $x=-1/z$. Then, $x=-1/z$ and $2x=2\times(-1/z)=-2/z$ so that $g(-1/z)=f(-1/z)-f(-2/z)$. $\endgroup$
    – parsiad
    Aug 30, 2016 at 19:45

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You blindly substitute the occurences of $z$ in the $g$ function WITH parenthesis around $z$ when the occurence is not trivial and then do some algebra to possibly simplify it. $$ g(-1/z) = f(-1/z) - f(2 (-1/z)) = f(-1/z) - f(-2/z) $$

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