Let f = {(0, –1), (–1, 3), (2, 3), (3, 5)} be a function from Z to Z defined by

Given: $f=\{(0,-1),(-1,-3),(2,3),(3,5)\}$ is a function from $Z$ to $Z$ defined by $f(x)=a x+b$

$f=\{(0,-1),(-1,-3),(2,3),(3,5)\}$ defined by $f(x)=a x+b$

$f(0)=-1$

$\Rightarrow a(0)+b=-1$

$\Rightarrow 0+b=-1$

$\Rightarrow b=-1$              ...(1)

$f(2)=3$

$\Rightarrow a(2)+b=3$

$\Rightarrow 2 a+b=3$

$\Rightarrow 2 a-1=3$        (From (1))

$\Rightarrow 2 a=3+1$

$\Rightarrow 2 a=4$   

$\Rightarrow a=2$      ...(2)

Thus,

$a=2$ and $b=-1$

Hence, $(a, b)=(\underline{2}-1)$.

Disclaimer: The function f must be equal to f  =  {(0, –1), (–1, –3), (2, 3), (3, 5)}.