Define two functions c(x) = max{x - a,0}, p(x) = max{a-x,0}. Prove that c(x) - p(x) = x - a

Question

Define two functions c(x) = max{x - a,0}, p(x) = max{a-x,0}. Prove that c(x) - p(x) = x - a

Answer 1

Fix an x. There are three possibilities. The first is that x-a>0. Then the function c(x)=x-a and p(x)=0 and hence, c(x)-p(x)=x-a. The second possibility is that x-a<0. Then the function c(x)=0 and p(x)=a-x and hence, c(x)-p(x)=0-(a-x)=x-a. The third possibility is that x-a=0 in which case we have c(x)-p(x)=0-0=0=x-a. Therefore, in all cases we have the equality c(x)-p(x)=x-a.