Use the bisection method to find the root of:
x = e^-x
Read "The Method" from here.
Bisection method - Wikipedia, the free encyclopedia
Choose $\displaystyle a=-1 $ and $\displaystyle b=1$ as endpoints for the first iteration
Now find c the midpoint that bisects the interal. $\displaystyle c = \frac{b+a}{2} = \frac{1-1}{2} = 0$
find $\displaystyle f(a), f(b)$ and $\displaystyle f(c)$
You will find
$\displaystyle f(a) = f(-1)<0 $
$\displaystyle f(b) = f(1) >0 $
$\displaystyle f(c) = f(0) <0 $
So now as $\displaystyle f(c) < 0$ choose it and $\displaystyle f(b) > 0$ to be the new interval as they have opposite signs as so on. The interval will get smaller and converge on the solution.