Question

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

Answer 1

Here f(x) = ax + b.

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

$\Rightarrow f\left(1\right)=1,f\left(2\right)=3,f\left(0\right)=-1,f(-1)=-3$

Now $f\left(1\right)=1\Rightarrow a\times 1+b=1\Rightarrow a+b=1$ ....(i)

$f\left(2\right)=3\Rightarrow a\times 2+b=3\Rightarrow 2a+b=3$ ....(ii)

Substracting (i) from (ii) we get

$2a+b-(a+b)=3-1\Rightarrow a=2$

Putting a = 2 in (i)

$2+b=1\Rightarrow b=-1.$