Answer;
Option B is correct.
The function increases at a constant multiplicative rate.
Step-by-step Explanation:
Complete Question
A table representing the function f(x) = 2(3/2)ˣ is shown below. What is true of the given function?
x | f(x)
0 | 2
1 | 3
2 | 4.5
3 | 6.75
a) The function increases at a constant additive rate.
b) The function increases at a constant multiplicative rate.
c) The function has an initial value of 0.
d) As each x value increases by 1, the y values increase by 1.
Solution
The function provided is
f(x) = 2(3/2)ˣ
We will take as h of the options one at a time to investigate which one is true.
a) The function increases at a constant additive rate.
A function that increases at an additive rate is one whose value as x increases, also changes with a constant difference between consecutive terms. That is, the consecutive terms of the functions for consecutive whole number values of x, will be like an Arithmetic progression with a constant common difference. If this function increased at an additive rate,
f(1) - f(0) = f(2) - f(1) = f(3) - f(2)
Inserting the values
3 - 2 = 4 5 - 3 = 6.75 - 4.5
1 ≠ 1.5 ≠ 2.25
Hence, this statement isn't true.
b) The function increases at a constant multiplicative rate.
For the function to increase at a multiplicative rate, the consecutive terms of the functions for consecutive whole number values of x will behave like a geometric progression with a common ratio. That is, the division of the (n+1)th term and the nth term of the function will always be a constant. That is,
f(1) ÷ f(0) = f(2) ÷ f(1) = f(3) ÷ f(2)
Inserting the values
(3/2) = (4.5/3) = (6 75/4.5)
1.5 = 1.5 = 1.5
The ratio is truly constant, hence, we can conclude that this statement is true and the function increases at a constant multiplicative rate.
c) The function has an initial value of 0.
For this to be true, f(0) = 0.
But from the table, it is evident that f(0) = 2, hence, the initial value of the function isn't 0 and this statement is false.
d) As each x value increases by 1, the y values increase by 1.
We've already checked the difference between consecutive terms of the functions for consecutive whole number values of x, the difference includes, 1, 1.5 and 2.25. Hence, as each x value increases by 1, the y values do NOT always increase by 1 and this statement is false.
Hope this Helps!!!
