Let f = {(1, 1), (2, 3), (0, -1), (-1, -3)} be a function from Z to Z defined by f(x) = ax + b, for some integers a, b.

Using (1, 1): 1 = a(1) + b ⇒ a + b = 1

Using (2, 3): 3 = a(2) + b ⇒ 2a + b = 3

Subtracting equations: (2a + b) - (a + b) = 3 - 1 ⇒ a = 2

Substitute a = 2 in a + b = 1 ⇒ 2 + b = 1 ⇒ b = -1

∴ a = 2, b = -1

∴ f(x) = 2x - 1