From the Intermediate Value Theorem, is it guaranteed that there is a root of the given equation in the given interval? cos(x)=x, (0,1) cos(0)= 1 cox(1)= 0.540... So using intermediate value theorem, no. and x=0 can't be possible because 0 was excluded in the domain by the round bracket so 0<x<1. Therefore there can't be a root. In other words the domain makes it so the graph is discontinuous at 0 (f(0) does not exist) and if the graph/function doesn't at that point there cannot be a root there. The answer is yes, but I am struggling to understand why and I am not sure if the books answer is wrong or if I'm wrong. *I also don't know why cos(x) = x , because when x=1 they aren't equal and would only be true at 1 point in this domain (when x=~0.79).