Question

Many doctors rely on the use of intravenous medication administration in order to achieve an immediate response of a particular drug's effects. The concentration, C, in mg/L, of a particular medication after being injected into a patient can be given by the function C of t is equal to the quantity negative 10 times t squared plus 50 times t end quantity over the quantity t squared plus 4 times t plus 3 end quantity comma where the time, t, is hours after injection.

What is the domain of the function C(t) based on the context of the problem?

Answer 1

The domain of the function representing the concentration of medication after intravenous injection is all non-negative values of time (t) starting from zero, because you cannot measure concentration before the medication is injected and the function's denominator has no positive real roots. Thus, the domain is [0, \infty).

Step-by-step explanation:

The domain of the function C(t) = {-10t^2 + 50t}/{t^2 + 4t + 3}, which represents the concentration of a medication after intravenous injection, is determined by considering values of t (time in hours after injection) for which the function is defined. In the context of the problem, time, t, must be a non-negative value because you cannot measure the medication concentration before it is administered.

Hence, the domain of C(t) starts from zero and extends to all positive values of t. However, we must also identify and exclude from the domain any values of t that would make the denominator equal to zero, as this would result in an undefined expression. The denominator t^2 + 4t + 3 factors to (t+1)(t+3). However, since the time after medication administration cannot be negative, the factors leading to negative time are not relevant, leaving the denominator with no real positive roots. Therefore, the domain based on the context of the problem is [0, \infty).