Let $f:[-1,1]\to[-1,1]$ be a strictly increasing such that $$f(-1)=-1,\quad\quad f(1)=1,\quad\quad f'(-1)=f'(1)=0$$
Letting $f$ be the cubic function uniquely characterized by these conditions, I don't think the inverse can be expressed without trigonometry or complex arithmetic:
$$2 \sin \left(\frac{1}{3} \sin ^{-1}(y)\right)=-\frac{i \left(-1+\left(\sqrt{1-y^2}+i y\right)^{2/3}\right)}{\sqrt[3]{\sqrt{1-y^2}+i y}}$$
Is there any alternative which (as well as its inverse) has a closed form consisting of only non-complex addition, subtraction, multiplication, division, and powers (preferably only square-root and integer powers)?
Ned's suggestion gave me the inspiration to realize this simple solution:
$\ x\mapsto x\sqrt{2-x^2}\ $ with the simple inverse $\ y\mapsto\sqrt{\frac{1+y}{2}}-\sqrt{\frac{1-y}{2}}$