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Section 1-1 : Review : Functions

31. Find the domain of \(\displaystyle A\left( x \right) = \frac{4}{{x - 9}} - \sqrt {{x^2} - 36} \).

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Hint : The domain of this function will be the set of all values of \(x\) that will work in both terms of this function.
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The domain of this function will be the set of all \(x\)’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,

\[x - 9 \ne 0\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\hspace{0.25in}x \ne 9\]

For the second term we need to require,

\[{x^2} - 36 \ge 0\hspace{0.25in}\,\,\, \to \hspace{0.25in}\,\,\,\,\,\,\,{x^2} \ge 36\hspace{0.25in}\,\,\,\, \Rightarrow \hspace{0.25in}\,\,\,\,\,\,x \le - 6\,\,\,\,\& \,\,\,\,x \ge 6\]
Hint : What values of \(x\) are in both of these?
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Now, we just need the set of \(x\)’s that are in both conditions above. In this case the second condition gives us most of the domain as it is the most restrictive. The first term is okay as long as we avoid \(x = 9\) and because this point will in fact satisfy the second condition we’ll need to make sure and exclude it. The domain is then,

\[{\mbox{Domain : }}x \le - 6\,\,\,\,\& \,\,\,\,x \ge 6,\,\,\,x \ne 9\]