Answer :
To determine the properties of the function \( f(x) = 2\left(\frac{3}{2}\right)^x \) from the given table, let's analyze the data systematically:
The table provided is:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 0 & 2 \\ \hline 1 & 3 \\ \hline 2 & 4.5 \\ \hline 3 & 6.75 \\ \hline \end{array} \][/tex]
### Step 1: Check for Constant Additive Rate
First, let's check if the function increases at a constant additive rate. We will look at the differences between consecutive \( y \)-values (\( f(x) \)):
[tex]\[ \begin{aligned} &f(1) - f(0) = 3 - 2 = 1, \\ &f(2) - f(1) = 4.5 - 3 = 1.5, \\ &f(3) - f(2) = 6.75 - 4.5 = 2.25. \end{aligned} \][/tex]
The differences are \( 1 \), \( 1.5 \), and \( 2.25 \), which are not constant. Therefore, the function does not increase at a constant additive rate.
### Step 2: Check for Constant Multiplicative Rate
Next, let's check if the function increases at a constant multiplicative rate. We will look at the ratios between consecutive \( y \)-values:
[tex]\[ \begin{aligned} &\frac{f(1)}{f(0)} = \frac{3}{2} = 1.5, \\ &\frac{f(2)}{f(1)} = \frac{4.5}{3} = 1.5, \\ &\frac{f(3)}{f(2)} = \frac{6.75}{4.5} = 1.5. \end{aligned} \][/tex]
The ratios are all \( 1.5 \), which indicates a constant multiplicative rate. Hence, the function increases at a constant multiplicative rate.
### Step 3: Initial Value
The initial value of the function when \( x = 0 \) is \( f(0) = 2 \), not \( 0 \). Therefore, the statement that the initial value is \( 0 \) is incorrect.
### Step 4: Increment of \( y \)-Values with \( x \)-Values
Finally, let's examine if each \( y \)-value increases by \( 1 \) as each \( x \)-value increases by \( 1 \):
From the differences calculated in Step 1:
- \( f(1) - f(0) = 1 \)
- \( f(2) - f(1) = 1.5 \)
- \( f(3) - f(2) = 2.25 \)
These differences are not equal to \( 1 \). Therefore, the statement that as each \( x \)-value increases by \( 1 \), the \( y \)-values increase by \( 1 \) is incorrect.
### Conclusion:
Among the given statements, the true statement about the function \( f(x) = 2\left(\frac{3}{2}\right)^x \) is that:
The function increases at a constant multiplicative rate.
The table provided is:
[tex]\[ \begin{array}{|c|c|} \hline x & f(x) \\ \hline 0 & 2 \\ \hline 1 & 3 \\ \hline 2 & 4.5 \\ \hline 3 & 6.75 \\ \hline \end{array} \][/tex]
### Step 1: Check for Constant Additive Rate
First, let's check if the function increases at a constant additive rate. We will look at the differences between consecutive \( y \)-values (\( f(x) \)):
[tex]\[ \begin{aligned} &f(1) - f(0) = 3 - 2 = 1, \\ &f(2) - f(1) = 4.5 - 3 = 1.5, \\ &f(3) - f(2) = 6.75 - 4.5 = 2.25. \end{aligned} \][/tex]
The differences are \( 1 \), \( 1.5 \), and \( 2.25 \), which are not constant. Therefore, the function does not increase at a constant additive rate.
### Step 2: Check for Constant Multiplicative Rate
Next, let's check if the function increases at a constant multiplicative rate. We will look at the ratios between consecutive \( y \)-values:
[tex]\[ \begin{aligned} &\frac{f(1)}{f(0)} = \frac{3}{2} = 1.5, \\ &\frac{f(2)}{f(1)} = \frac{4.5}{3} = 1.5, \\ &\frac{f(3)}{f(2)} = \frac{6.75}{4.5} = 1.5. \end{aligned} \][/tex]
The ratios are all \( 1.5 \), which indicates a constant multiplicative rate. Hence, the function increases at a constant multiplicative rate.
### Step 3: Initial Value
The initial value of the function when \( x = 0 \) is \( f(0) = 2 \), not \( 0 \). Therefore, the statement that the initial value is \( 0 \) is incorrect.
### Step 4: Increment of \( y \)-Values with \( x \)-Values
Finally, let's examine if each \( y \)-value increases by \( 1 \) as each \( x \)-value increases by \( 1 \):
From the differences calculated in Step 1:
- \( f(1) - f(0) = 1 \)
- \( f(2) - f(1) = 1.5 \)
- \( f(3) - f(2) = 2.25 \)
These differences are not equal to \( 1 \). Therefore, the statement that as each \( x \)-value increases by \( 1 \), the \( y \)-values increase by \( 1 \) is incorrect.
### Conclusion:
Among the given statements, the true statement about the function \( f(x) = 2\left(\frac{3}{2}\right)^x \) is that:
The function increases at a constant multiplicative rate.
