Question

If the function $f:\{-1,1\}\to \{0,2\}$, defined by $f\left(x\right)=ax+b$ is a surjection, then find $a$ and $b$.

Answer 1

We have

$f\left(x\right)=ax+b$

Case-I

$f(-1)=2$ $b-a=2$

$f\left(1\right)=2$ $b+a=2$

So,

$a=b=1$

Case-II

$f(-1)=2$ $b-a=2$

$f\left(1\right)=0$ $b+a=0$

So

$(-1,1)$

Then

$a=-1$ and $b=1$

Which is the required answer.