Question

Let f : W $\to$ W be defined as

f(n)=$$\{\begin{array}{c}n-1,ifnisodd\\ n+1,ifniseven\end{array}$$
Show that f is invertible and find the inverse of f. Here, W is the set of all whole numbers.

Answer 1

$f:W\to W$

$f\left(n\right)$ $=\{n-1,n=oddn+1,n=even\}$

When $n$ is odd

$f\left(n_1\right)=f\left(n_2\right)$

$n_1-1=n_2-1$

$n_1=n_2$

When $n$ is even

$f\left(n_1\right)=f\left(n_2\right)$

$n_1+1=n_2+1$

$n_1=n_2$

So, $f\left(n\right)$ is one-one

When $n$ is odd

$f\left(n\right)=n-1$

$y=n-1$

$n=y+1$

Put $n$ in $f\left(n\right)$

$f\left(n\right)=y+1-1$

$f\left(n\right)=y$

When $n$ is even

$f\left(n\right)$=n+1

$y=n+1$

$n=y-1$

Put $n$ in $f\left(n\right)$

$f\left(n\right)=y-1+1$

$f\left(n\right)=y$

So, $f\left(n\right)$ is onto

So, the function $f\left(n\right)$ is bijective. Hence is invertible.

$f\left(n\right)=n-1$ if $n$ is odd

$y=n-1$

$n=y+1$

$f^{-1}\left(n\right)=y+1$ if $n$ is odd

$f\left(n\right)=n+1$ if $n$ is even

$y=n+1$

$n=y-1$

$f^{-1}\left(n\right)=y-1$ if $n$ is even