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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 $$t = 9$$.