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 whole numbers.

if n is odd

if n is even.

For one-one:

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.

Onto:

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 ,

and

Hence, f is inertible  and inverse of f is g i.e. , which is same as f.

Hence, inverse of f is f itself.

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