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I have a basic question about the notation for writing functions. Say that I have a function defined as

$$f(x) = g(x) - 1.$$

Is it then correct to write

$$ f = g -1?$$

Are there any problems with writing functions like this?

Aurantium
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    It is clearly amboguous: $f$ and $g$ can stay also for numbers... – Mauro ALLEGRANZA Jun 20 '17 at 13:13
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    @MauroALLEGRANZA: If it were clearly ambiguous to this person, they wouldn't be asking the question. – Moya Jun 20 '17 at 17:17
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    Just a note if you're trying this in school: even though it's correct for a mathematician, don't be surprised if you get points off for it. Some instructors will take points off for anything they haven't taught you in class, some might be worried another student who doesn't know the meaning of this will try to complain why you got the points and they didn't, etc... – user541686 Jun 21 '17 at 03:57

2 Answers2

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If you use $f$ as the symbol for some function, then $f(x)$ is the image of $x$ under that function, i.e. $f$ represents a function and $f(x)$ is a number, the result of $x \mapsto f(x)$. So it makes sense to write $f(x)-1$ as a function value where $1$ is subtracted from $f(x)$: there's no real risk on wrong interpretations here, this is standard notation.

However, you can also do operations on functions such as the sum of functions $f+g$ or the product of functions $fg$. In that context, you could interpret "$f-1$" as a difference of functions: the difference of $f$ and a constant function $x \mapsto 1$ (for all $x$), abbreviated simply as "$1$".

Note then that the blue $1$ in $f(x)-\color{blue}{1}$, a number, doesn't play the same role as the red $1$ in $f-\color{red}{1}$, meant to denote a function. This can raise confusion so only do this when it's sufficiently clear from the context what you mean.

StackTD
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  • I don't understand where the confusion comes from. You can't subtract a number from a function and you can't subtract a function from a number. So it is clear that 1 is a number in f(x)-1 and 1 is a function in f-1. On the other hand, I find f(x)=g(x)-1 confusing (even if it is of course standard notation), because you don't know if you are defining f or if you are trying to solve for x. – FCardelle May 10 '21 at 16:02
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In $f(x) = g(x)-1$, the $1$ is a number. In $f = g - 1$, the $1$ is a function $1(x)$, the function that for any value of $x$ will always return the constant $1$. So long as you and whoever is reading your work know that difference, you should be okay to write it.

Carsten S
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Bob Krueger
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    One problem with this is that $1$, in a function role, might in some contexts (namely, linear functions) be construed to denote the identity function, rather than the constant-1 function. – leftaroundabout Jun 21 '17 at 15:01
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    @leftaroundabout, thanks for the very good note. I've never really seen $1$ refer to the constant-1 function, but that was what seemed to be implied in the OP's question. Normally, $1$, when used as a function, refers to some identity function (I've seen $1_A$, for example), but I have not seen this in the context of functions of a real variable. The statement "So long as you and whoever is reading your work knows the difference" applies in any case, however. – Bob Krueger Jun 21 '17 at 15:33