Let `f: W ->W`be defined as `f(n) = n - 1`, if is odd and `f(n) = n + 1`, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole numbers.

Explanation

In `f: W rarr W`
`{:{(n-1 ", If n is odd. "),(n+1 ", If n is even. "):}`
Let f(n) = f(m) (if n is odd and m is even )
`rArr n-1 =m-1 `
`rArr n=m `
and if n and m both are even then
f(n) = f(m)
`rArr n+1 =m+1`
`rArr n+m therefore` is one- one
Now ,each even number 2r in co-domain W is the image of odd number 2r+1 in co- domain W is the image of even number 2r in the domain W.
`therefore` is onto
`therefore` f is one- one onto
`therefore` is invertible
`therefore` g, the inverse of f is defined as :
` g: W rarr W`
`g(m)= {{:("m+1, if m is even "),("m-1 , if m is odd "):}`