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Relations and Functions




7. Composition of Functions:

Let f : A → B and g : B → C be two functions. Then the composition of f and g, denoted by gof, is defined as the function gof : A -> C given by
gof=g(f(x) for all x ∈ A

Example
f(x) =(x+3)
g(x) =x2

gof=g(f(x))=g(x+3)=(x+3)2
Similarly
fog=f(g(x))=f(x2)=x2 +3

In this case
fog ≠ gof

8. Invertible Function:

If the Function f : A-> B is both one to one and onto i.e bijective ,then we can find a function g: B-> A
such that
g(y)=x when y=f(x). It is denoted as f-1. The function f(x) is called invertible function

Another defination of Invertible function A Function f : A-> B is invertible if we can find a function g: B- > A such that
fog=y gof=x
Example
A set A is defined as A={a,b,c}
Let f: A-> A be the function defined as are
1) f={(a,a),(b,b),(c,c)}
2) f={(a,b),(b,a),(c,c)}
3) f={(a,c),(b,c),(c,a)}
Find if all these function defined are invertible
Solutions 1) The neccesary condition for invertibleness is one on one and onto
This function is clearly one on one and onto,so it is invertible
2) This function is clearly one on one and onto,so it is invertible
3) This function is not one on one and neither onto,so it is not invertible


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