It can be seen from these results that the value of f (x) changes from −4 at f (2) to +6 at f (4), indicating that a root lies between 2 and 4. This is shown more clearly in Fig. 9.1.
As shown above, by using functional notation it is possible to determine the vicinity of a root of an equation by the occurrence of a change of sign, i.e. if x1 and x2 are such that f (x1) and f (x2) have opposite signs, there is at least one root of the equation f (x) = 0 in the interval between x1 and x2 (provided f (x) is a continuous function). In the method of bisection the mid-point of the interval, i.e. x3 = x1 + x22 , is taken, and from the sign of f (x3) it can be deduced whether a root lies in the half interval to the left or right of x3. Whichever half interval is indicated, its mid-point is then taken and the procedure repeated. The method often requires many iterations and is therefore slow, but never fails to eventually produce the root. The procedure stops when two successive value of x are equal-to the required degree of accuracy.