Why do you use F(x) in one place and f(x) in another?

I'll assume you mean that f(x) is defined as

f(0) = 3

f(1) = 2

f(2) = 1

f(3) = 0

f(4) = -1

f(5) = -1

and we want to find 1/f(x).

Let g(x) = 1/f(x).

It's just

g(0) = 1/3

g(1) = 1/2

g(2) = 1

g(3) = undefined

g(4) = -1

g(5) = -1