Question

The number of fish in a fish tank doubles each week. The function y=equals 3 (2)^x represents the population, where X is the number of one week periodsa. Describe the domains and range of the function. Then graph the functionb. Find and interpret the y intercept c. How many fish are in the tank at the end of first week?d. How many fish are in the tank after 4 weeks?

Answer 1

a. The domain of the function is the set of values of the independent variable (x) on which the function acts, in this case, the independent variable is time, since the time can only take positive values, the domain would be [0,∞) or x≥0.

The range is all the possible values that the dependent variable can take, in this case, the dependent variable is the population of fish, then it can't be a negative number, since you can't have for example -1 fish, then, again, the interval would be [0,∞) or y≥0

A graph of the function looks like this:

b. We can find the y-intercept by making x equals 0 in the formula of the function, like this:

y= 3*(2)^x

y= 3*(2)^0

y= 3*1, since any number raised to zero equals 1

y=3

Since x equals zero represents the initial time, the y-intercept represents the initial population of fish, then at the beginning, there were 3 fish.

c. We just have to replace 1 for x, and then calculate y, like this:

y= 3*(2)^x

y= 3*(2)^1

y= 3*(2)

y=6

At the end of the first week, there will be 6 fish.

d. Now, we just have to replace 4 for x, like this:

y= 3*(2)^4

y= 3*16

y=48

Then, after 4 weeks, there will be 48 fish