Bracketing roots consists of narrowing down the region in which a root of a one-dimensional equation may exist into an interval. The basic principle of bracketing involves the intermediate value theorem which states that if you have a continuous function f(x), and f(a) is a different sign than f(b), then somewhere in between there is a root. Below is a diagram which will give an initutive feel for why this must be so. To see a formal statement and proof of the IVT press here. In practical numerical work, there are several circumstance in which the initial value theorem will fail you. The first case is a jump discontinuity across zero.