Chapter 1 - Precalculus Review - 1.3 The Basic Classes of Functions - Exercises - Page 23: 43

See proofs

Compute $\delta (f+g)(x)$: $\delta (f+g)(x)=\delta (f(x)+g(x))=(f(x+1)+g(x+1))-(f(x)+g(x))$ $=f(x+1)+g(x+1)-f(x)-g(x)$ $=(f(x+1)-f(x))+(g(x+1)-g(x))$ $=\delta f(x)+\delta g(x)$ We got: $\delta (f+g)(x)=\delta f(x)+\delta g(x)$ $\delta (f+g)=\delta f+\delta g$ Compute $\delta (cf)(x)$: $\delta (cf)(x)=(cf)(x+1)-(cf)(x)$ $=cf(x+1)-cf(x)$ $=c(f(x+1)-f(x))$ $=c\delta f(x)$ We got: $\delta (cf)(x)=c\delta f(x)$ $\delta (cf)=c\delta (f)$