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I'm in what seems the first year of college (I'm in France so it doesn't have the same name)

In calculus and other math classes I came across multiple meaning/definition for a single notation. I know the context of the exercise and teacher preferences help know which definition to use but I'd like to know what the official notation dictates.

Here are the definitions I'm talking about:

  • $f^n(x) = [f(x)]^n$ e.g.: $\sin^2(x)$;
  • $f^n(x) = f \circ f \circ f \circ \cdots \circ f (x)$;
  • $f^{-1}(y)=x$ when $f(x)=y$ and $f$ is bijective;
  • $f^{(n)}(x) = \frac{d^nf}{dx^n}$
Sebastiano
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DrinRchd
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    There is no "official" notation in mathematics. Everything depends on convention and context. – K.defaoite Nov 08 '23 at 20:36
  • Remark that if $f$ is bijective, the $n$-repeated compositions $f^n$ and $f^{-1}$ can be seen as the same notation. Usually one would write $f(x)^n$ if it is exponentiation of the number $f(x)$. With indeed the notable exception of $\sin.$ But it should be clear from context. I would count the $n$-th derivative as a different notation as it uses the brackets $(\cdot)$. Again, it should always be clear from context ... – Neckverse Herdman Nov 08 '23 at 20:39
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    Now you know why many texts/people write $\arcsin x$ instead of $\sin^{-1}x$. – Ted Shifrin Nov 08 '23 at 20:57

2 Answers2

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The meaning of a particular notation depends on the context. If your teacher or the author of some paper defines $f^n(x) = \frac{d^n}{dx^n} f(x)$ to be the $n$th derivative of $f(x)$, then that is exactly what it means in that context. There is rarely, if ever, a universal or "official" meaning assigned to a given notation.

For this reason, it never hurts to define exactly what you mean when using a particular notation. However, it's not always necessary to do so, especially if your audience already has an agreed upon understanding of what it means.

RyRy the Fly Guy
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  • Thanks. Should I consider expliciting the meaning of the notation when I use it then? – DrinRchd Nov 08 '23 at 20:46
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    Knowing French math teachers, you either should explicitly state what your notation entails, or use unambiguous notation – Zadig Nov 08 '23 at 20:47
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    It never hurts to define exactly what you mean when using a particular notation. However, it's not always necessary to do so, especially if your audience already has an agreed upon understanding of what it means. If your teachers demand that you do so on your assignments, then you should do it. – RyRy the Fly Guy Nov 08 '23 at 20:49
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Note that you actually only stumbled upon two different meanings for the same notation.

Firstly, $f^{(n)}$ is standard notation for the $n$-th derivative. Anyone in a calculus or analysis context with some familiarity of the field will understand it. The brackets make it a different notation from $f^{n}$, so it's not a new meaning for the same notation.

Now the notation $f^n$ usually refers to repeated application of a function. In this context, function composition $\circ$ is viewed as an operation just like $\cdot$ or $+$, and the notation for repeatedly doing it is copied from multiplication. Also note that the notation $f^{-1}$ extends this in exactly the same way as it does for multiplication. $2^{-n}$ is the number which, if multiplied by $2^n$, gives the multiplicative identity $1$. The function $f^{-n}$ is the function which, if composed with $f^n$, gives the identity function.

However, in some calculus or analysis contexts, it can also mean normal exponentiation. For instance, it is rare to have a useful application for a function like $\sin(\sin(x))$, but there are useful contexts for products of sines, so in context everyone understands that $\sin^2$ is $\sin\cdot\sin$, not $\sin\circ\sin$.

With time you will get a better feeling for when each meaning applies. But usually it is clear from context - unless you are not yet familiar enough with the field, or the author communicates badly.

Vercassivelaunos
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