F(x) is an exponential function with these features: f(x) is increasing, the y-intercept is (0,1), and f(2)=49. Which function could be f(x)?
f(x)=3x
f(x)=(17)x
f(x)=7x
f(x)=(13)x
The total municipal debt for the state of Illinois can be represented as the exponential function M(t)=41.24(1.052)t, where M represents the total municipal debt for the state in billions of dollars and t is the number of years since 2000. Determine the statement that interprets the function M(t).(1 point)
The total municipal debt in Illinois was $41.24 million in 2000 and increases about 1.052% each year.
The total municipal debt in Illinois was $43.38 billion in 2000 and increases about 5.2% each year.
The total municipal debt in Illinois was $41.24 billion in 2000 and increases about 5.2% each year.
The total municipal debt in Illinois was $39.20 billion in 2000 and increases about 105.2% each year.
A newspaper for a large city launches a new advertising campaign focusing on the number of digital subscriptions. The equation S(t)=31,500(1.034)t approximates the number of digital subscriptions S as a function of t months after the launch of the advertising campaign. Determine the statements that interpret the parameters of the function S(t).
Select the two correct answers.
(1 point)
The newspaper had 32,571 digital subscriptions when the advertising campaign was launched.
The newspaper had 31,500 digital subscriptions when the advertising campaign was launched.
The number of digital subscriptions increases by 103.4% each month.
The number of digital subscriptions increases by 3.4% each year.
The number of digital subscriptions increases by 3.4% each month.
My answers
1.b
2.c
3.B,D
1. For the first question, the correct answer is f(x)=(13)x. The exponential function that is increasing and passes through the point (0,1) has the form f(x)=(a)x, where a is a positive number. Since f(2)=49, we can substitute the values into the equation and solve for a: 49=(a)^2. Solving for a, we get a=7 or a=-7. However, since we are given that f(x) is increasing and the y-intercept is (0,1), we can discard the negative value and conclude that f(x)=(13)x.
2. For the second question, the correct interpretation of the function M(t) is: The total municipal debt in Illinois was $41.24 billion in 2000 and increases about 5.2% each year. This is because the initial value, 41.24, represents the total municipal debt in billions of dollars in 2000, and the growth factor, 1.052, represents the increase in debt each year.
3. For the third question, the correct interpretations of the parameters of the function S(t) are:
- The newspaper had 31,500 digital subscriptions when the advertising campaign was launched.
- The number of digital subscriptions increases by 3.4% each month.
This is because the initial value, 31,500, represents the number of digital subscriptions when the advertising campaign was launched, and the growth factor, 1.034, represents the monthly increase in subscriptions.
1. The correct answer is f(x)=(13)x.
Since the y-intercept is (0,1), we can substitute x=0 into the function to get f(0)=13^0=1. This matches the y-intercept.
Since f(2)=49, we can substitute x=2 into the function to get f(2)=13^2=169. This does not match f(2)=49, so the correct answer is not f(x)=(17)x or f(x)=3x. The correct answer is f(x)=(13)x.
2. The correct interpretation of the function M(t)=41.24(1.052)t is:
The total municipal debt in Illinois was $41.24 billion in 2000 and increases about 5.2% each year.
3. The correct interpretations of the parameters of the function S(t)=31,500(1.034)t are:
- The newspaper had 31,500 digital subscriptions when the advertising campaign was launched.
- The number of digital subscriptions increases by 3.4% each month.
For the first question, we are given an exponential function that is increasing, has a y-intercept of (0,1), and f(2)=49. To determine which function could be f(x), we need to find the equation that satisfies these conditions.
Let's analyze each option:
a) f(x) = 3x
This is a linear function, not an exponential function. Therefore, it does not meet the given conditions.
b) f(x) = (17)x
This is an exponential function, but it does not have a y-intercept of (0,1). Therefore, it does not meet the given conditions.
c) f(x) = 7x
This is a linear function, not an exponential function. Therefore, it does not meet the given conditions.
d) f(x) = (13)x
This is an exponential function and it has a y-intercept of (0,1). Let's check if f(2) = 49:
f(2) = (13)^2 = 169, which is not equal to 49. Therefore, it does not meet the given conditions.
So, the correct function for f(x) is not listed among the options.
For the second question, we are given the function M(t) = 41.24(1.052)^t, which represents the total municipal debt for the state of Illinois in billions of dollars as a function of the number of years since 2000. We need to determine the statement that interprets this function correctly.
Analyzing the options:
a) The total municipal debt in Illinois was $41.24 million in 2000 and increases about 1.052% each year.
This statement is incorrect because the function is in billions of dollars, not millions.
b) The total municipal debt in Illinois was $43.38 billion in 2000 and increases about 5.2% each year.
This statement is incorrect because the initial debt in 2000 is given as $41.24 billion, not $43.38 billion.
c) The total municipal debt in Illinois was $41.24 billion in 2000 and increases about 5.2% each year.
This statement is correct. The initial debt in 2000 is $41.24 billion, and the function shows an annual increase of approximately 5.2%.
d) The total municipal debt in Illinois was $39.20 billion in 2000 and increases about 105.2% each year.
This statement is incorrect because the increase in the function is approximately 5.2%, not 105.2%.
So, the correct interpretation is statement c) The total municipal debt in Illinois was $41.24 billion in 2000 and increases about 5.2% each year.
For the third question, we are given the function S(t) = 31,500(1.034)^t, which represents the number of digital subscriptions for a newspaper as a function of t months after the advertising campaign is launched. We need to determine the statements that interpret the parameters of this function correctly.
Analyzing the options:
a) The newspaper had 32,571 digital subscriptions when the advertising campaign was launched.
This statement is incorrect because the initial number of digital subscriptions is given as 31,500, not 32,571.
b) The newspaper had 31,500 digital subscriptions when the advertising campaign was launched.
This statement is correct. The initial number of digital subscriptions is given as 31,500.
c) The number of digital subscriptions increases by 103.4% each month.
This statement is incorrect because the increase in the function is approximately 3.4%, not 103.4%.
d) The number of digital subscriptions increases by 3.4% each year.
This statement is incorrect because the function represents the number of digital subscriptions as a function of months, not years.
So, the correct interpretations are statements b) The newspaper had 31,500 digital subscriptions when the advertising campaign was launched, and e) The number of digital subscriptions increases by 3.4% each month.
Therefore, the correct answers are:
1. None of the options (f(x) does not match any given function)
2. c) The total municipal debt in Illinois was $41.24 billion in 2000 and increases about 5.2% each year.
3. b) The newspaper had 31,500 digital subscriptions when the advertising campaign was launched. and e) The number of digital subscriptions increases by 3.4% each month.