Compositions Of Functions Compositions Of Functions Function is defined as a relationship or connection between a set of inputs and outputs. It also supports the statement that for each input there exists some particular output. If ‘a’ is input to a function ‘f ’ then there must exist a variable, as an output for this input a. every function has a set of domain and a set of range. Domain refers to the set of inputs which are used in the function or which can be putted into the function to obtain a particular output. And range is defined as a set in which output is obtained on inserting or substituting inputs. composition of functions uses the concept of domain and range in an important executive manner. As composition is performed using these two parameters. Composition of functions basically refers to the combination of two functions. That is it is defined as using one function into a second function. Which means range of first function becomes the domain for second function.

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Composition of two functions cannot be always commutative. In some cases it can be commutative. Commutative means, like addition of two numbers follows commutative property, as it produces the same output independent of, in which way it is performed. a+ b or b+ a, both gives same output. Consider two function, say f(x) and g(x) where value of f(x) = 2x and value of g(x) = 3x +1 now, suppose we need to find the values of f (1) and g (1). F (1) gives 2 on substituting 1 in f(x) and g(1) gives 4 on substituting 1 in g(x). Now if we want to check whether the functions are commutative or not, den we need to find f(g(x)) and g(f(x)), they must be equal. f(g(x)) gives 8 on substituting value of g(x) into f(x) and g(f(x)) gives 7 on substituting value of f(x) into g(x). Since, values of both functions is unequal, therefore these functions are not commutative. Thus composition of functions involves substitution or use of one function into another function. F (g(x)) and g (f(x)) gives an idea of composition of functions. Tree diagrams are defined as the diagrams which are represented in the form of trees and branches. Every tree contains one parent node and many child nodes. A tree consists of different number of nodes. These nodes are known as child nodes. Basically use of tree diagrams in mathematics is used to represent all possible outcomes or solutions for one problem or for one question. Thus, tree diagrams make the solution easier to understand, it is also used to find the optimum solution which can be obtained for a particular problem or question. Composition monoids :- Suppose one has two (or more) functions f: X → X, g: X → X having the same domain and codomain. Then one can form long, potentially complicated chains of these functions composed together, such as f ∘ f ∘ g ∘ f.

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Such long chains have the algebraic structure of a monoid, called transformation monoid or composition monoid. In general, composition monoids can have remarkably complicated structure. One particular notable example is the de Rham curve. The set of all functions f: X → X is called the full transformation semigroup on X. If the functions are bijective, then the set of all possible combinations of these functions forms a transformation group; and one says that the group is generated by these functions. The set of all bijective functions f: X → X form a group with respect to the composition operator. This is the symmetric group, also sometimes called the composition group. Alternative notations :- Many mathematicians omit the composition symbol, writing gf for g ∘ f. In the mid-20th century, some mathematicians decided that writing "g ∘ f" to mean "first apply f, then apply g" was too confusing and decided to change notations. They write "xf" for "f(x)" and "(xf)g" for "g(f(x))". This can be more natural and seem simpler than writing functions on the left in some areas – in linear algebra, for instance, when x is a row vector and f and g denote matrices and the composition is by matrix multiplication. This alternative notation is called postfix notation. The order is important because matrix multiplication is non-commutative. Successive transformations applying and composing to the right agrees with the left-to-right reading sequence.

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