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


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



$f(x)=a x+b$

$(1,1) \in f$

$\Rightarrow f(1)=1$

$\Rightarrow a \times 1+b=1$

$\Rightarrow a+b=1$

$(0,-1) \in f$

$\Rightarrow f(0)=-1$

$\Rightarrow a \times 0+b=-1$

$\Rightarrow b=-1$

On substituting $b=-1$ in $a+b=1$, we obtain $a+(-1)=1 \Rightarrow a=1+1=2$.

Thus, the respective values of $a$ and $b$ are 2 and $-1$.

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