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I got a question from a friend, who is tutoring high school students for Math after school. The question is basically this:

Does writing $f(x) = x^2$ mathematically mean the same as $F(x) = x^2$?

I read from other S.E sections that function notations are conventions, not fixed rules. How will my friend explain to her students about when to (mathematically correct) use notations $f(x)$, $g(x)$, $h(x)$ or $F(x)$, $G(x)$, $H(x)$?

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    $\begingroup$ There is no universal convention about what type of letters to use... You have to note that, as per the above definition, for every value assigned to $x$ the two functions $f(x)$ and $F(x)$ have the same value; thus, basically, they are the same "object". $\endgroup$ Sep 11, 2017 at 6:26
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    $\begingroup$ You can denote a function with any letter or even a non-letter symbol and it would be "mathematically correct". There are some conventions within specific branch of mathematics. E.g. when you teach about integrals, you would often use big $F$ for an antiderivative of small $f$ and in topology open sets are most often denoted with $U,V,W$ instead of $A,B,C$. But it's just for convenience, if you defined it in a different way it would be also be correct, even if less common. $\endgroup$
    – Joanna F
    Sep 11, 2017 at 6:33
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    $\begingroup$ You can draw a squirrel if you want, and use that as the name of your function. It would be a hassle, and trying to type it on a computer would really be troublesome compared to the alternatives, but it wouldn't be uncorrect. Just uncommon. $\endgroup$
    – Arthur
    Sep 11, 2017 at 7:22

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No, they mean different things, at least to one of your dumb computers. Try defining $f(x) = x^2$ in a Mathematica notebook, then later in the same notebook, ask for the value of $F(\phi)$. It probably won't answer $\phi + 1$, not unless you have also defined $F(x) = x^2$. Most likely it will complain that $F$ is undefined, or maybe you have defined that for something else.

Humans are only slightly smarter than computers. A human sees $F(5)$ and he might think you mean either the fifth Fibonacci number or the fifth Fermat number.

So, for clarity, I suggest you use $f(x)$ for the first generic function you define in a given context, and $F(x)$ for a function named after some man or monster, god or demon. Or don't follow this suggestion, I will enjoy the ensuing confusion. Mwahahahahahahahaha!

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