section_path: Iterated functions
section_name: Iterated functions
paragraph_text: Iterated functions
  
  Function composition is a binary operation that is defined on functions such that the codomain of the function written on the right is included in the domain of the function written on the left. It is denoted g ∘ f, {\displaystyle g\circ f,} and defined as
  
  for every x in the domain of f.
  
  If the domain of a function f equals its codomain, one may compose the function with itself an arbitrary number of time, and this defines the n th power of the function under composition, commonly called the n th iterate of the function. Thus f n {\displaystyle f^{n}} denotes generally the n th iterate of f; for example, f 3 (x) {\displaystyle f^{3}(x)} means f (f (f (x))). {\displaystyle f(f(f(x))).}
  
  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 f 2 (x) = f (f (x)), {\displaystyle f^{2}(x)=f(f(x)),} and f (x) 2 = f (x) ⋅ f (x). {\displaystyle f(x)^{2}=f(x)\cdot f(x).} When functional notation is not used, disambiguation is often done by placing the composition symbol before the exponent; for example f ∘ 3 = f ∘ f ∘ f, {\displaystyle f^{\circ 3}=f\circ f\circ f,} and f 3 = f ⋅ f ⋅ f. {\displaystyle f^{3}=f\cdot f\cdot f.} For historical reasons, the exponent of a repeated multiplication is placed before the argument for some specific functions, typically the trigonometric functions. So, sin 2 x {\displaystyle \sin ^{2}x} and sin 2 (x) {\displaystyle \sin ^{2}(x)} both mean sin (x) ⋅ sin (x) {\displaystyle \sin(x)\cdot \sin(x)} and not sin (sin (x)), {\displaystyle \sin(\sin(x)),} which, in any case, is rarely considered. Historically, several variants of these notations were used by different authors.
  
  In this context, the exponent − 1 {\displaystyle -1} denotes always the inverse function, if it exists. So sin − 1 x = sin − 1 (x) = arcsin x. {\displaystyle \sin ^{-1}x=\sin ^{-1}(x)=\arcsin x.} For the multiplicative inverse fractions are generally used as in 1 / sin (x) = 1 sin x. {\displaystyle 1/\sin(x)={\frac {1}{\sin x}}.}