Question

y=f(x)=1/x. Use algebra to find a simplified rational expression for the slope of the line between (3, f(3)) and (3+h, f(3+h))?? hcannot=0.

Answer 1

\[f(x) = \frac{1}{x}\]
\begin{align*}
\large \implies \left. \frac{\Delta f(x)}{\Delta x} \right|_{x \in [3,3+h]} &= \frac{\frac{1}{3+h} - \frac{1}{3}}{(3+h) - 3} \\
&= \frac{\frac{1}{3+h} - \frac{1}{3}}{(3+h) - 3}\cdot \frac{3(3+h)}{3(3+h)}\\
&= \frac{3 - (3+h)}{3h(3+h)}\\
&= \frac{3-3-h}{3h(3+h)} \\
&= \frac{-\cancel{h}}{3\cancel{h}(3+h)} \\
&= -\frac{1}{9+3h}
\end{align*}