Composite versus square notation in functions

Question

What is meaning of $f^2(x)$ ? There seems to be confusion in its interpretation.

Is $f^2(x)$ same as $(f(x))^2$ or $f\circ f$?

Answer 1

It depends upon the context. It can be both. But if $f$ is a function from some set $S$ into $\mathbb R$, then it can only be $\bigl(f(x)\bigr)^2$. And if it is a function from a set $S$ into itself and if that set has no multiplication defined in it, then it can only mean $f\circ f$.

Answer 2

It can be both, but unless you are in a context where function iteration makes sense, it is more likely to be the square.

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The confusion is a little worse with the exponent $-1$: $\sin^{-1}(x)$ is more likely to be the arc sine than the cosecant. The functional inverse is not so exotic.

Answer 3

I would interpret it as $(f(x))^2$ whenever it makes sense. If $f(x)$ is not defined (for example, if it is a vector), then I would understand $f\circ f$

Anyway, I would make sure the meaning is consistent throughout my work, never mixing them.

In topics such as linear algebra, I will always mean $f \circ f$ If I am talking about calculus, I would expect you to understand $(f(x))^2$