Consider f N Z defined as fn 1n2n114 This function is bij

Consider f : N -> Z defined as f(n) = ((-1)^n(2n-1)+1)/4. This function is bijective. Find its inverse

Solution

y= ((-1)^n(2n-1)+1)/4

for finding inverse, replace n with y and y with n.

n = ((-1)^y(2y-1)+1)/4

4n-1 = (-1)^y(2y-1)

if y is odd, then (-1)^y = -1

4n-1 = -2y + 1

y = 1-2n(inverse)

if y = even , (-1)^2 = 1

4n-1 = 2y-1

y = 2n.(inverse)