Let $f\left(x\right)$ be a linear function which maps $\left[-1,1\right]$  onto $\left[0,2\right]$ ,then $f\left(x\right)$ can be

$\Rightarrow 0<f\left(x\right)<2$

Range of $f$ is equal to co domain$\therefore f\left(x\right)=x+1$  

$-1<x<1\Rightarrow 1>-x>-1$

$\Rightarrow -1<-x<1$

$\Rightarrow 0<-x+1<2$

$\Rightarrow 0<f\left(x\right)<2$

Range of $f$ is equal to co domain $\therefore f\left(x\right)=-x+1$