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Section 3-4 : The Definition of a Function

16. Determine the domain of the following function.

\[g\left( t \right) = \frac{{8t - 24}}{{{t^2} - 7t - 18}}\] Show Solution

Recall that the domain is simply all the values of \(t\) that we can plug into the function and the resulting function value will be a real number.

In this case we have a rational expression where both the numerator and denominator are polynomials.

The numerator won’t cause any problems since we can plug any value of \(t\) into the numerator and get back a real number.

So, for this problem, all we need to avoid is division by zero. We will get division by zero at,

\[{t^2} - 7t - 18 = 0\hspace{0.25in} \to \hspace{0.25in} \left( {t - 9} \right)\left( {t + 2} \right) = 0 \hspace{0.25in} \to \hspace{0.25in} t = - 2,\,\,\,\,t = 9\]

Therefore, the domain for this function is all real numbers except \(t = - 2\) and \(t = 9\).