Dont really understand how to do this problem.
Use the Intermediate Value Theorem to show that there is a root of the equation in the given interval.
e^(-x^2)) = x (0,1)
Let $\displaystyle f(x)=e^{-x^2}-x$, observe that $\displaystyle f$ is a continuous function. Since $\displaystyle f(0)=1$ and $\displaystyle f(1)<0$,(because $\displaystyle e>1\Rightarrow e^{-1}<1 \Rightarrow e^{-1}-1<0$) thanks to IVT, $\displaystyle f(x)=0$ in $\displaystyle (0,1)\Rightarrow e^{-x^2}=x$ there.