Question

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

Answer 1

Step 1: Given:

$f=\left\{\right(1,1),(2,3),(0,-1),(-1,3\left)\right\}$
$f\left(x\right)=ax+b$

Step 2: Find the function

The given points satisfy the function.
For points $(1,1):$
$f\left(1\right)=1$
$\Rightarrow a+b=1$ ....(1)

For point $(0,-1)$ :
$f\left(0\right)=-1$
$\Rightarrow a.0+b=-1$
$\Rightarrow b=-1$ ....(2)

From equation (1) and (2) we get.
$\Rightarrow a-1=1$
$\Rightarrow a=2\text{and}b=-1$
Substituting the value of $aandb$ in given function
$\Rightarrow f\left(x\right)=2x-1$