The number of linear functions which map $\left[-1,1\right]$ to $\left[0,2\right]$ are

Let the required function be $f\left(x\right)=ax+b$ 

$\text{if \hspace{0.17em}}f(-1)=0\text{and}f\left(1\right)=2$
$\Rightarrow -a+b=0\text{and}a+b=2$
$\Rightarrow a=1\text{and}b=1$
$\text{if \hspace{0.17em}}f(-1)=2\text{and}f\left(1\right)=0$
$\Rightarrow -a+b=2\text{and}a+b=0$
$\Rightarrow a=-1\text{and}b=1$
Hence $f\left(x\right)=x+1orf\left(x\right)=-x+1$