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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,

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