## 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) =x

^{2}
gof=g(f(x))=g(x+3)=(x+3)

^{2}
Similarly

fog=f(g(x))=f(x

^{2})=x

^{2} +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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