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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.
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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?
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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,

\[{\mbox{Domain : }} - \frac{9}{2} \le y < 2\]