Let's analyze the given function and the data.
The table represents the function:
[tex]\[ f(x) = 2 \left( \frac{3}{2} \right)^x \][/tex]
Here’s the given data:
[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 have multiple statements to evaluate:
1. The function increases at a constant additive rate:
- This means the difference between consecutive [tex]\( f(x) \)[/tex] values should be constant.
- Checking the differences:
[tex]\[
\begin{align*}
f(1) - f(0) &= 3 - 2 = 1, \\
f(2) - f(1) &= 4.5 - 3 = 1.5, \\
f(3) - f(2) &= 6.75 - 4.5 = 2.25.
\end{align*}
\][/tex]
- The differences (1, 1.5, 2.25) are not constant.
- This statement is false.
2. The function increases at a constant multiplicative rate:
- This means the ratio of consecutive [tex]\( f(x) \)[/tex] values should be constant.
- Checking the ratios:
[tex]\[
\begin{align*}
\frac{f(1)}{f(0)} &= \frac{3}{2} = 1.5, \\
\frac{f(2)}{f(1)} &= \frac{4.5}{3} = 1.5, \\
\frac{f(3)}{f(2)} &= \frac{6.75}{4.5} = 1.5.
\end{align*}
\][/tex]
- The ratios (all 1.5) are constant.
- This statement is true.
3. The function has an initial value of 0:
- The initial value [tex]\( f(0) \)[/tex] is given by the function when [tex]\( x = 0 \)[/tex].
- According to the table, [tex]\( f(0) = 2 \)[/tex].
- This statement is false.
4. As each [tex]\( x \)[/tex] value increases by 1, the [tex]\( y \)[/tex] values increase by 1:
- This means the difference between consecutive [tex]\( f(x) \)[/tex] values should be exactly 1.
- Previously, we calculated the differences (1, 1.5, 2.25) which are not all 1.
- This statement is false.
Based on the analysis, the true statement about the given function is:
- The function increases at a constant multiplicative rate.
