Let $h\left(x\right)=g\left(f\left(x\right)\right)$ where, $g\left(x\right)$ is injective and $h\left(x\right)$ is surjective. $h\left(x\right)$ is surjective.

$\therefore$ Codomain of $h\left(x\right)=$ Range of $h\left(x\right)$

Range of $h\left(x\right)=\left[0,2\right]$ which is also codomain of $g.$ So, must be surjective.

Now, domain of $g=\left[-1,1\right]$ which must be range of $f.$ But codomain of $f=\left[-1,1\right]$

So, it must be surjective.