Chapter 5.2, Problem 66ES

Let $F$ and $G$ be the functions defined by

$F\left(x\right)=\frac{x^2+3x}{x}\text{ and }G\left(x\right)=\left\{\begin{array}{cc}x+3,& x>0\\ x,& x<0\end{array}\right.$

(a) Show that $F$ and $G$ have the same derivative.

(b) Show that $G\left(x\right)\ne F\left(x\right)+C$ for any constant $C$ .

(c) Do parts (a) and (b) contradict Theorem $\mathrm{5.2.2}$ ? Explain.