Editing Exponentiation (section)

==Iterated functions==
{{See also|Iterated function}}
[[Function composition]] is a [[binary operation]] that is defined on [[function (mathematics)|functions]] such that the [[codomain]] of the function written on the right is included in the [[domain of a function|domain]] of the function written on the left. It is denoted <math>g\circ f,</math> and defined as
: <math>(g\circ f)(x)=g(f(x))</math>
for every {{mvar|x}} in the domain of {{mvar|f}}.

If the domain of a function {{mvar|f}} equals its codomain, one may compose the function with itself an arbitrary number of time, and this defines the {{mvar|n}}th power of the function under composition, commonly called the ''{{mvar|n}}th iterate'' of the function. Thus <math>f^n</math> denotes generally the {{mvar|n}}th iterate of {{mvar|f}}; for example, <math>f^3(x)</math> means <math>f(f(f(x))).</math><ref name="Peano_1903"/>

When a multiplication is defined on the codomain of the function, this defines a multiplication on functions, the [[pointwise multiplication]], which induces another exponentiation. When using [[functional notation]], the two kinds of exponentiation are generally distinguished by placing the exponent of the functional iteration ''before'' the parentheses enclosing the arguments of the function, and placing the exponent of pointwise multiplication ''after'' the parentheses. Thus <math>f^2(x)= f(f(x)),</math> and <math>f(x)^2= f(x)\cdot f(x).</math> When functional notation is not used, disambiguation is often done by placing the composition symbol before the exponent; for example <math>f^{\circ 3}=f\circ f \circ f,</math> and <math>f^3=f\cdot f\cdot f.</math> For historical reasons, the exponent of a repeated multiplication is placed before the argument for some specific functions, typically the [[trigonometric functions]]. So, <math>\sin^2 x</math> and <math>\sin^2(x)</math> both mean <math>\sin(x)\cdot\sin(x)</math> and not <math>\sin(\sin(x)),</math> which, in any case, is rarely considered. Historically, several variants of these notations were used by different authors.<ref name="Herschel_1813"/><ref name="Herschel_1820"/><ref name="Cajori_1929"/>

In this context, the exponent <math>-1</math> denotes always the [[inverse function]], if it exists. So <math>\sin^{-1}x=\sin^{-1}(x) = \arcsin x.</math> For the [[multiplicative inverse]] fractions are generally used as in <math>1/\sin(x)=\frac 1{\sin x}.</math>