Q. 2 Let be defined as , if n is odd and , if n is
even. Show that f is invertible. Find the inverse of f. Here, W is the set of all
if n is odd
if n is even.
Taking x as odd number and y as even number.
Now, Taking y as odd number and x as even number.
This is also impossible.
If both x and y are odd :
If both x and y are even :
f is one-one.
Any odd number 2r+1 in co domain of N is image of 2r in domain N and any even number 2r in codomain N is image of 2r+1 in domain N.
Thus, f is onto.
Hence, f is one-one and onto i.e. it is invertible.
Sice, f is invertible.
Let as if m is even and if m is odd.
When x is odd.
When x is even
Similarly, m is odd
m is even ,
Hence, f is inertible and inverse of f is g i.e. , which is same as f.
Hence, inverse of f is f itself.