Iterated function different from function composition
In mathematics, an iterated function is a function X → X (that is, a function from some set X to itself) which is obtained  by composing another function f : X → X with itself a certain number of times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial object, the result of applying a given function is fed again in the function as input, and this process is repeated. For example on the image on the right:(f   
  
    
      
        ∘
      
    
    {\displaystyle \circ }
     g)(x) = f (g(x)),always associative.
Because the notation f n may refer to both iteration (composition) of the function f or exponentiation of the function f (the latter is commonly used in trigonometry), some mathematicians choose to use ∘ to denote the compositional meaning, writing f∘n(x) for the n-th iterate of the function f(x), as in, for example, f∘3(x) meaning f(f(f(x))). For the same purpose, f [n](x) was used by Benjamin Peirce whereas Alfred Pringsheim and Jules Molk suggested nf(x) instead.