The population f(x), in millions, of State A of a country after x years is represented by the function shown below:
f(x) = 4(1.08)x
The graph shows the population g(x), in millions, of State B of the country after x years:
graph of exponential function g of x that curves up from left to right and goes through points 0 comma 2 and 9 comma 4
Which conclusion is correct about the populations of State A and State B?
The original population of State A was double the original population of State B.
The original population of State B was double the original population of State A.
The original population of State A was four times the original population of State B.
The original population of State A was equal to the original population of State B.
The conclusion that is correct about the populations of State A and State B is: The original population of State A was four times the original population of State B.
This can be determined by comparing the initial values of both functions. The given function for State A is f(x) = 4(1.08)^x, which means that when x=0, the population of State A is 4(1.08)^0 = 4(1) = 4 million.
For State B, the graph shows that when x=0, the population of State B is 2 million.
Therefore, the original population of State A (4 million) is four times the original population of State B (2 million).
The function f(x) = 3,267(1 + 0.02)x represents the amount of money in a savings account where x represents time in years. What does the 3,267 represent?
The amount of money in the savings account after one year
The amount of money added to the savings account each year
The initial amount of money placed in the savings account
The growth rate
The 3,267 represents the initial amount of money placed in the savings account.
In the given function f(x) = 3,267(1 + 0.02)^x, the number outside the parentheses, 3,267, is the initial amount of money placed in the savings account.
Two different businesses model their profits over 15 years, where x is the year, f(x) is the profits of a garden shop, and g(x) is the profits of a construction materials business. Use the data to determine which function is exponential, and use the table to justify your answer.
x f(x) g(x)
1995 $14,170.20 $11,008.31
2000 $19,396.20 $16,174.82
2005 $24,622.20 $23,766.11
2006 $25,667.40 $25,667.40
2007 $26,712.60 $27,720.79
2010 $31,938.60 $34,920.21
g(x) is exponential; an exponential function increases faster than a linear function.
g(x) is exponential; g(x) increased less overall than f(x).
f(x) is exponential; an exponential function increases more slowly than a linear function.
f(x) is exponential; f(x) increased less overall than g(x).
To determine which function is exponential, we need to analyze the rate at which the function increases over time.
Looking at the given data, we can observe that g(x) demonstrates a consistent and significant increase in profits over the years, while f(x) shows a slower and less consistent increase. This behavior suggests that g(x) is exponential, and f(x) is not.
Therefore, the correct answer is: g(x) is exponential; an exponential function increases faster than a linear function.
Use the graph representing bacteria decay to estimate the domain of the function and solve for the average rate of change across the domain.
An exponential function titled Bacteria Decay with x axis labeled Time, in Minutes, and y axis labeled Amount of Bacteria, in Thousands, decreasing to the right with a y intercept of 0 comma 90 and an x intercept of 30 comma 0.
0 ≤ y ≤ 90, −0.33
0 ≤ y ≤ 90, −3
0 ≤ x ≤ 30, −0.33
0 ≤ x ≤ 30, −3
The correct answer is: 0 ≤ x ≤ 30, −0.33.
From the given graph, we can determine that the domain of the function is 0 ≤ x ≤ 30, which represents the range of possible values for the time in minutes.
To find the average rate of change, we need to calculate the slope of the line connecting the points (0, 90) and (30, 0). The average rate of change is given by the formula:
Average rate of change = (change in y) / (change in x)
The change in y is 0 - 90 = -90, and the change in x is 30 - 0 = 30.
Thus, the average rate of change is -90/30 = -3.
Therefore, the correct answer is: 0 ≤ x ≤ 30, -3.
Given the exponential function f(x) = 16(0.75)x, classify the function as exponential growth or decay and determine the percent rate of growth or decay.
Exponential growth, 75% increase
Exponential decay, 75% decrease
Exponential growth, 25% increase
Exponential decay, 25% decrease
Given the exponential function f(x) = 16(0.75)^x, we can determine its classification and the percent rate of growth or decay.
To determine if it is exponential growth or decay, we need to examine the base of the exponent (0.75) in the function. If the base is greater than 1, it represents exponential growth, and if it is between 0 and 1, it represents exponential decay.
In this case, the base of the exponent is 0.75, which is between 0 and 1. Therefore, the function represents exponential decay.
To find the percent rate of growth or decay, we can subtract the base (0.75) from 1 and express it as a percentage.
1 - 0.75 = 0.25
Therefore, the percent rate of decay is 25% decrease.
Hence, the correct answer is: Exponential decay, 25% decrease.
A ball is dropped from a certain height. The function below represents the height f(n), in feet, to which the ball bounces at the nth bounce:
f(n) = 9(0.7)n
What does the number 9 in the function represent?
Number of times the ball was dropped
Height from which the ball was dropped
Height to which the ball will bounce at the nth bounce
Number of times the ball bounces at the nth bounce