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Section 1.1 : Review : Functions

25. Find the domain of \(\displaystyle g\left( t \right) = \frac{{6t - {t^3}}}{{7 - t - 4{t^2}}}\).

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In 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,

\[7 - t - 4{t^2} = 0\hspace{0.25in}\hspace{0.25in} \Rightarrow \hspace{0.25in}\hspace{0.25in}t = \frac{{1 \pm \sqrt {{{\left( { - 1} \right)}^2} - 4\left( { - 4} \right)\left( 7 \right)} }}{{2\left( { - 4} \right)}} = - \frac{1}{8}\left( {1 \pm \sqrt {113} } \right)\]

The two values above are the only values of \(t\) that we can’t plug into the function. All other values of \(t\) can be plugged into the function and will return real values. The domain is then,

\[{\mbox{Domain : All real numbers except }}t = - \frac{1}{8}\left( {1 \pm \sqrt {113} } \right)\]