Which statements are true of functions? check all that apply.
a. all functions have a dependent variable.
b. all functions have an independent variable.
c. the range of a function includes its domain.
d. a vertical line is an example of a functional relationship.
e. a horizontal line is an example of a functional relationship.
f. each output value of a function can correspond to only one input value.

The biologist has a second culture to examine. she knows that the population of the culture doubles every 15 minutes. after 1 hour and 15 minutes, her assistant found that 80,000 bacteria were present. a) what was the size of the initial population? b) predict the size of the culture at t=3 hours. what was the size of the population at 40 minutes? explain and justify your answers. c) create a graph of the population as a function of time. find an equation that can be used to predict the size of the population at any time t. d) examine the rate at which the bacteria culture is growing. how fast is the culture growing after 1 hour? after 1.5 hours? after 2 hours? use a time interval of h = 0.01 hours to estimate these rates. interpret these rates in terms of the context of the problem situation. how do these three rates compare?