Does the function f of x equals 3 times the quantity 1 over 2 end quantity to the x power represent growth or decay? What is the y-intercept of f(x)?
Growth; 0 comma one-half
Growth; (0, 3)
Decay; 0 comma one-half
Decay; (0, 3)
The function f(x) = 3(1/2)^x represents decay because the base of the exponent is less than 1.
To find the y-intercept, we substitute x = 0 into the equation:
f(0) = 3(1/2)^0 = 3(1) = 3.
Therefore, the y-intercept of f(x) is (0, 3).
So the correct answer is Decay; (0, 3).
A bacteria is growing by a factor of 2 every hour from 1 p.m. to 11 p.m. The function below shows the number of bacterial cells, f(x), after x hours from 1 p.m.:
f(x) = 20(2)x
Which of the following is a reasonable domain for the function?
All positive integers
1 ≤ x ≤ 11
0 ≤ x ≤ 20
0 ≤ x ≤ 10
The function f(x) represents the number of bacterial cells after x hours from 1 p.m., where x is the number of hours.
Since the bacteria is growing from 1 p.m. to 11 p.m., it is reasonable to assume that the domain for the function is from 0 to 10 (inclusive), as it represents the number of hours within this time period.
So the correct answer is 0 ≤ x ≤ 10.
Which of the following options results in a graph that shows exponential decay?
f(x) = 0.6(2)x
f(x) = 3(0.7)x
f(x) = 0.4(1.6)x
f(x) = 20(3)x
Exponential decay occurs when the base of the exponent is between 0 and 1.
Looking at the given options:
- Option f(x) = 0.6(2)^x has a base of 2, which is greater than 1, so it does not represent decay.
- Option f(x) = 3(0.7)^x has a base of 0.7, which is between 0 and 1, so it represents decay.
- Option f(x) = 0.4(1.6)^x has a base of 1.6, which is greater than 1, so it does not represent decay.
- Option f(x) = 20(3)^x has a base of 3, which is greater than 1, so it does not represent decay.
Therefore, the option that results in a graph showing exponential decay is f(x) = 3(0.7)^x.
Finbar invested money in a transportation stock whose growth is modeled by the function f(x) = 0.01(2)x, where x represents number of days. Find the approximate average rate of change from day 11 to day 15.
0.013
4
76.8
307.2
To find the average rate of change from day 11 to day 15, we need to find the difference in the function values at these two points and divide it by the difference in the days.
Substituting x = 11 into the function f(x) = 0.01(2)^x:
f(11) = 0.01(2)^11 ≈ 0.0224
Substituting x = 15 into the function:
f(15) = 0.01(2)^15 ≈ 0.032
The difference in the function values is:
f(15) - f(11) ≈ 0.032 - 0.0224 ≈ 0.0096
The difference in the number of days is 15 - 11 = 4.
Therefore, the approximate average rate of change from day 11 to day 15 is:
0.0096 / 4 ≈ 0.0024.
So the correct answer is 0.0024, which is approximately 0.013 when rounded.