College Algebra, Chapter 3, 3.4, Section 3.4, Problem 18

A function $\displaystyle g(x) = \frac{2}{x + 1}$. Determine the average rate of change of the function between $x = 0$ and $x = h$.


$\begin{equation} \begin{aligned} \text{average rate of change } =& \frac{g(b) - g(a)}{b - a} && \text{Model} \\ \\ \text{average rate of change } =& \frac{g(h) - g(0)}{h - 0} && \text{Substitute } a = 0 \text{ and } b = h \\ \\ \text{average rate of change } =& \frac{\displaystyle \frac{2}{h + 1} - \frac{2}{0 + 1} }{h} && \text{Simplify} \\ \\ \text{average rate of change } =& \frac{\displaystyle \frac{2}{h + 1} - 2}{h} && \text{Get the LCD} \\ \\ \text{average rate of change } =& \frac{2 - 2 (h + 1)}{h(h + 1)} && \text{Apply Distributive Property} \\ \\ \text{average rate of change } =& \frac{2 - 2h - 2}{h (h + 1)} && \text{Combine like terms} \\ \\ \text{average rate of change } =& \frac{-2 \cancel{h}}{\cancel{h} (h + 1)} && \text{Cancel out like terms} \\ \\ \text{average rate of change } =& \frac{2}{(h + 1)} && \text{Answer} \end{aligned} \end{equation}$