Question

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

Answer 1

$f=\left\{\right(1,1),(2,3),(0,-1),(-1,-3\left)\right\}$
$f\left(x\right)=ax+b$
$(1,1)\in f$
$\Rightarrow f\left(1\right)=1$
$\Rightarrow a\times 1+b=1$
$\Rightarrow a+b=1$ .....(1)
$(0,-1)\in f$
$\Rightarrow f\left(0\right)=-1$
$\Rightarrow a\times 0+b=-1$
$\Rightarrow b=-1$
On substituting $b=-1$ in eqn (1), we get
$a+(-1)=1$
$\Rightarrow a=1+1=2$
Thus the respective values of $a$ and $b$ are $2$ and $-1$