Answer:
1a. increasing. It is increasing for all values of t.
1b. no. The population asymptotically approaches 40,000.
2. the sign of the slope of a reciprocal function is opposite the sign of the function's slope
Explanation:
Given the population function P(t) = 20(4t+3)/(2t+5), you want to know if the population is increasing or decreasing, and if the population will reach 50,000. You want to have an explanation of why the reciprocal of an increasing linear function is decreasing.
1. Population
The population function can be expanded to ...
P(t) = 40 -140/(2t+5)
a. Slope
As we know from question 2, the second term is increasing, hence the population function is increasing. (The basic reciprocal function is decreasing, but it is subtracted here, so the overall effect is an increasing function.)
b. Maximum
The magnitude of the second term of the above version of the population function starts at 28 for t=0 and decreases asymptotically to zero. Hence the population starts at 12000 and increases to an asymptote of 40,000. It will never reach 50,000.
2. Reciprocal function
The basic reciprocal function is f(x) = 1/x. It is decreasing everywhere it is defined. It is the reciprocal of a linear function with positive slope (an increasing function).
Translation or vertical or horizontal scaling of the basic function (using positive scale factors) does not change the sign of the slope, either of the original linear function or of its reciprocal. Hence the reciprocal of an increasing function is decreasing.
Looking at derivatives, if f'(x) is positive, then the derivative of g(x) = 1/f(x) is ...
g'(x) = (-1/f(x)^2)f'(x)
That is, f(x)^2 is non-negative, and -f'(x) is negative, so the derivative g'(x) must be negative (wherever f(x)≠0). The reciprocal function is decreasing.