College Algebra, Chapter 5, 5.1, Section 5.1, Problem 46

The hyperbolic sine function is defined by
$\displaystyle \sin h(x) = \frac{e^x - e^{-x}}{2}$
a.) Draw the graph of the functions $\displaystyle y = \frac{e^x}{2} \text{ and } y = \frac{-e^{-x}}{2}$
On the same axes, and use graphical addition to sketch the graph of $y = \sin h (x)$
The graph of $y = \sin h (x)$



Using graphical addition, the graph of $y = \sin h (x)$ is



b.) Show that $\sin h(-x) = -\sin h(x)$ using the definition

$
\begin{equation}
\begin{aligned}
\sin (-x) &= \frac{e^{-x}-e^{-(-x)}}{2}, -\sin h (x) = - \left( \frac{e^x - e^{-x}}{2} \right)\\
\\
\sin (-x) &= \frac{e^{-x}-e^x}{2} \qquad \text{Factor out } -1\\
\\
\sin (-x) &= -\left( \frac{e^x - e^{-x}}{2} \right) = -\sin h (x)
\end{aligned}
\end{equation}
$