Let's analyze the function [tex]\( f(x) = 2\left(\frac{3}{2}\right)^x \)[/tex] using the given information.
First, we will fill in the missing value in the table and then check the properties of the function.
1. Initial Value Calculation:
When [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 2\left(\frac{3}{2}\right)^0 = 2 \cdot 1 = 2
\][/tex]
Therefore, the initial value [tex]\( i \)[/tex] at [tex]\( x = 0 \)[/tex] is 2.
The completed table is:
[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]
2. Checking Additive Growth:
Let's calculate the differences between consecutive [tex]\( f(x) \)[/tex] 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]
The differences are 1, 1.5, and 2.25, so the function does not increase by a constant additive rate. Therefore, the function does not meet the condition of increasing at a constant additive rate.
3. Checking Multiplicative Growth:
Let's calculate the ratios between consecutive [tex]\( f(x) \)[/tex] 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]
The ratios are all equal to 1.5, so the function increases by a constant multiplicative rate. Therefore, the function does meet the condition of increasing at a constant multiplicative rate.
4. Initial Value Condition:
The initial value of the function [tex]\( f(x) \)[/tex] is 2, not 0. Therefore, this statement is false.
5. Checking the Y Value Increment as X Increases by 1:
As calculated in step 2, the differences (increments) [tex]\( f(x) \)[/tex] are 1, 1.5, and 2.25, which means the [tex]\( y \)[/tex] values do not increase by 1 when [tex]\( x \)[/tex] increases by 1. Thus, this statement is false.
Summary:
- The function increases at a constant additive rate. (False)
- The function increases at a constant multiplicative rate. (True)
- The function has an initial value of 0. (False)
- As each [tex]\( x \)[/tex] value increases by 1, the [tex]\( y \)[/tex] values increase by 1. (False)
So, the only correct statement about the function is that it increases at a constant multiplicative rate.
