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

\end{aligned}
\end{equation}
$