How does the location of a power in a function adjust its result?

For example if $f(x) = 2x$

I assume that $f(x^2)$ would equal $2x^2$.

What I am unsure about is if $f^2(x)$ means $f(x)\times f(x)$ or $f(f(x))$. So would the final answer be $(2x)(2x)=4x^2$ or $2(2x)=4x$.

And is $f(x)^2$ the same as $f^2(x)$?

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    $\begingroup$ generally its the latter one $\endgroup$ – Archis Welankar Feb 24 '18 at 18:05
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    $\begingroup$ I'd say that $f^2(x)$ was highly ambiguous and I would expect context to clarify. I agree that $f\circ f$ would be the more common meaning, but I think it should be spelled out. $\endgroup$ – lulu Feb 24 '18 at 18:05
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    $\begingroup$ You may find the answers here interesting math.stackexchange.com/a/2663269/504810 $\endgroup$ – Air Conditioner Feb 24 '18 at 18:08
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    $\begingroup$ Indeed there is an ambiguity. To avoid such confusion some uses the notation $f^{\circ n}$ to denote the $n$-fold iterated function. $\endgroup$ – Sangchul Lee Feb 24 '18 at 18:12
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    $\begingroup$ When $f(x)^2$ is used, you can safely assume that it means $(f(x))^2$. $f(x^2)$ is always written as $f(x^2)$. $\endgroup$ – user228113 Feb 24 '18 at 18:19

You are right about $f(x^2)$; and $f(x)^2$ simply means $f(x)f(x)$. The troublesome notation is $f^2$, whether used to talk about the function $f^2$ or its value at the point $x$, namely $f^2(x)$. One convention often used is that it is the function whose values are the squared values of $f$, namely $f^2:x\mapsto f(x)^2$. However, in the sort of mathematics where we are more interested in compounding functions than in doing algebraic operations on their values, concatenation is a convenient way to represent composition, particularly because such composition is associative. Thus, $fgh$ means $x\mapsto f(g(h(x)))$. Using a more explicit expression of composition, we might write this associativity as $(f\circ g)\circ h=f\circ(g\circ h)$, but writing all those little circles is rather tedious when we are not doing algebra with the values and don't need to distinguish it from composition.

In any text where there is a need to do both compounding of functions and algebraic operations on their values to a significant extent, the authors will normally be aware of the problem and carefully explain just what notational conventions they are using.

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  • $\begingroup$ I think in abstract algebra it is pretty common for $f^2$ to indicate $f\circ f$ as well, as it is in line with how we denote other binary operations applied multiple times and also extends to negatives to denote the inverse functions $\endgroup$ – aidangallagher4 Apr 3 '18 at 10:22
  • $\begingroup$ The moral is: if you use notation $f^2(x)$, then define what you mean. In every new document, define it again. If that is too inconvenient for you, then use $f(x)^2$ or $f^{\circ 2}(x)$, or ... Exception: trig functions, but even though $\sin^2 x$ and $\sin^{-1} x$ have known meanings, still students are confused by them all the time. $\endgroup$ – GEdgar Apr 3 '18 at 13:16

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