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Section 1-1 : Review : Functions

26. Find the domain of \(g\left( x \right) = \sqrt {25 - {x^2}} \).

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In this case we need to avoid square roots of negative numbers so we need to require,

\[25 - {x^2} \ge 0\]

Note that once we have the original inequality written down we can do a little rewriting of things as follows to make things a little easier to see.

\[{x^2} \le 25\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\,\,\,\,\, - 5 \le x \le 5\]

At this point it should be pretty easy to find the values of \(x\) that will keep the quantity under the radical positive or zero so we won’t need to do a number line or sign table to determine the range.

The domain is then,

\[{\mbox{Domain : }} - 5 \le x \le 5\]