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
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
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
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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