Respuesta :
General Idea:
Domain of a function means the values of x which will give a DEFINED output for the function.
Applying the concept:
Given that the x represent the time in seconds, f(x) represent the height of food packet.
Time cannot be a negative value, so
[tex] x\geq 0 [/tex]
The height of the food packet cannot be a negative value, so
[tex] f(x)\geq 0 [/tex]
We need to replace [tex] -15x^2+6000 [/tex] for f(x) in the above inequality to find the domain.
[tex] -15x^2+6000\geq 0 \; \; [Divide \; by\; -15\; on\; both\; sides]\\ \\ \frac{-15x^2}{-15} +\frac{6000}{-15} \leq \frac{0}{-15} \\ \\ x^2-400\leq 0\;[Factoring\;on\;left\;side]\\ \\ (x+200)(x-200)\leq 0 [/tex]
The possible solutions of the above inequality are given by the intervals [tex] (-\infty , -2], [-2,2], [2,\infty ) [/tex]. We need to pick test point from each possible solution interval and check whether that test point make the inequality [tex] (x+200)(x-200)\leq 0 [/tex] true. Only the test point from the solution interval [-200, 200] make the inequality true.
The values of x which will make the above inequality TRUE is [tex] -200\leq x\leq 200 [/tex]
But we already know x should be positive, because time cannot be negative.
Conclusion:
Domain of the given function is [tex] 0\leq x\leq 200 [/tex]
