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Section 1.1 : Review : Functions
30. Find the domain of \(\displaystyle h\left( y \right) = \sqrt {2y + 9} - \frac{1}{{\sqrt {2 - y} }}\).
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Hint : The domain of this function will be the set of all values of \(y\) that will work in both terms of this function.
The domain of this function will be the set of all \(y\)’s that we can plug into both terms in this function and get a real number back as a value. This means that we first need to determine the domain of each of the two terms.
For the first term we need to require,
\[2y + 9 \ge 0\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\hspace{0.25in}y \ge - \frac{9}{2}\]For the second term we need to require,
\[2 - y > 0\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\hspace{0.25in}y < 2\]Note that we need the second condition to be strictly positive to avoid division by zero as well.
Hint : What values of \(y\) are in both of these?
Now, we just need the set of \(y\)’s that are in both conditions above. In this case we need all the \(y\)’s that will be greater than or equal to \( - \frac{9}{2}\) AND less than 2. The domain is then,
\[{\mbox{Domain : }} - \frac{9}{2} \le y < 2\]