To determine what is true about the given function, we need to analyze the values presented in the table and relate them to the function \( f(x) = 2 \left( \frac{3}{2} \right)^x \).
Given table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
x & f(x) \\
\hline
0 & 2 \\
\hline
1 & 3 \\
\hline
2 & 4.5 \\
\hline
3 & 6.75 \\
\hline
\end{tabular}
\][/tex]
We will check each of the possible statements systematically:
1. The function increases at a constant additive rate:
To check if the function increases at an additive rate, we should compare the differences between consecutive \( f(x) \) values.
[tex]\[
f(1) - f(0) = 3 - 2 = 1
\][/tex]
[tex]\[
f(2) - f(1) = 4.5 - 3 = 1.5
\][/tex]
[tex]\[
f(3) - f(2) = 6.75 - 4.5 = 2.25
\][/tex]
Since the differences are not the same (1, 1.5, 2.25), the function does not increase at a constant additive rate.
2. The function increases at a constant multiplicative rate:
To check if the function increases at a multiplicative rate, we should compare the ratios between consecutive \( f(x) \) values:
[tex]\[
\frac{f(1)}{f(0)} = \frac{3}{2} = 1.5
\][/tex]
[tex]\[
\frac{f(2)}{f(1)} = \frac{4.5}{3} = 1.5
\][/tex]
[tex]\[
\frac{f(3)}{f(2)} = \frac{6.75}{4.5} = 1.5
\][/tex]
Since all the ratios are equal (1.5), the function increases at a constant multiplicative rate.
3. The function has an initial value of 0:
The initial value of the function is \( f(0) = 2 \), not 0. Therefore, this statement is false.
4. As each \( x \) value increases by 1, the \( y \) values increase by 1:
We have already calculated the differences between consecutive \( f(x) \) values (1, 1.5, 2.25), and none of them all equal 1. Therefore, this statement is false.
Given the analysis above:
- The statement "The function increases at a constant multiplicative rate" is true.
Thus, the correct conclusion about the function [tex]\( f(x) = 2 \left( \frac{3}{2} \right)^x \)[/tex] is that it increases at a constant multiplicative rate.
