Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best viewed in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (you should be able to scroll/swipe to see them) and some of the menu items will be cut off due to the narrow screen width.
Section 1.1 : Review : Functions
24. Find the domain of \(\displaystyle R\left( z \right) = \frac{5}{{{z^3} + 10{z^2} + 9z}}\).
Show SolutionIn this case we need to avoid division by zero issues so we’ll need to determine where the denominator is zero. To do this we will solve,
\[{z^3} + 10{z^2} + 9z = z\left( {{z^2} + 10z + 9} \right) = z\left( {z + 1} \right)\left( {z + 9} \right) = 0\hspace{0.25in}\,\,\,\, \Rightarrow \hspace{0.25in}z = 0,\,\,z = - 1,\,\,z = - 9\]The three values above are the only values of \(z\) that we can’t plug into the function. All other values of \(z\) can be plugged into the function and will return real values. The domain is then,
\[{\mbox{Domain : All real numbers except }}z = 0,\,\,\,z = - 1,\,\,z = - 9\]