Question:
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 $a, b$.
Solution:
$f=\{(1,1),(2,3),(0,-1),(-1,-3)\}$
$f(x)=a x+b$
$(1,1) \in f$
$\Rightarrow f(1)=1$
$\Rightarrow a \times 1+b=1$
$\Rightarrow a+b=1$
$(0,-1) \in f$
$\Rightarrow f(0)=-1$
$\Rightarrow a \times 0+b=-1$
$\Rightarrow b=-1$
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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