One element of a related set is assigned to each element of the set by using a Function. For representing a computational complexity of algorithms, for counting objects, for studying the sequences and strings, and for naming some of them, functions are used.
The relationship from the elements of one set X to elements of another set Y is defined as function or mapping, which is represented as f:X→Y. For the function ‘f’, X is the domain or pre-image and Y is the codomain of image. Function ‘f’ is a relation on X and Y such that for each x∈X, there exists a unique y∈Y such that (x,y)∈R. A function can be one to one or many to one.
A function f:A→B is injective or one-to-one function if for every b∈B, there exists at most one a∈A such that f(s)=t.
This means a function f is injective if
A function f:A→B is surjective (onto) if the image of f equals its range. Equivalently, for every b∈B, there exists some a∈A such that f(a)=b. This means that for any y in B, there exists some x in A such that y=f(x).
If a function f is both injective and surjective, then the function f:A→B is bijective or one-to-one correspondent.
Prove that a function f:R→R defined by f(x)=2x–3 is a bijective function.
Explanation – This function is proved by using both injective and surjective functions.
So, x=(y+5)/3 which belongs to R and f(x)=y.
Hence, f is surjective.
Since f is both surjective and injective, it is said tha f is bijective.
The inverse of a one-to-one corresponding function f:A→B, is the function g:B→A, holding the following property −
The function f is called invertible, if its inverse function g exists.
Two functions f:A→B and g:B→C can be composed to give a composition gof. This is a function from A to C defined by (gof)(x)=g(f(x))
Let f(x)=x+2 and g(x)=2x+1, find (fog)(x) and (gof)(x).
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