Questions: A table representing the function (f(x)=2left(frac32right)^x) is shown below. x f(x) 0 2 1 3 2 4.5 3 6.75 What is true of the given function? - The function increases at a constant additive rate. - The function increases at a constant multiplicative rate. - The function has an initial value of 0. - As each (x) value increases by 1, the (y) values increase by 1.

$A table representing the function (f(x)=2left(frac32right)^x) is shown below. x f(x) 0 2 1 3 2 4.5 3 6.75 What is true of the given function? - The function increases at a constant additive rate. - The function increases at a constant multiplicative rate. - The function has an initial value of 0. - As each (x) value increases by 1, the (y) values increase by 1.$

Transcript text: 2.corelearn.edgenuity.com/player/ \& Measurement $A$ Assignment Active Practice with exponential growth functions. A table representing the function $f(x)=2\left(\frac{3}{2}\right)^{x}$ is shown below. \begin{tabular}{|c|c|} \hline $\boldsymbol{x}$ & $\boldsymbol{f}(\boldsymbol{x})$ \\ \hline 0 & 2 \\ \hline 1 & 3 \\ \hline 2 & 4.5 \\ \hline 3 & 6.75 \\ \hline \end{tabular} What is true of the given function? The function increases at a constant additive rate. The function increases at a constant multiplicative rate. The function has an initial value of 0 . As each $x$ value increases by 1 , the $y$ values increase by 1 .

Solution

Solution Steps

Step 1: Analyze the Function

The given function is $ f(x) = 2\left(\frac{3}{2}\right)^{x} $. This is an exponential function, where the base of the exponent is $\frac{3}{2}$.

Step 2: Determine the Type of Growth

Exponential functions increase at a constant multiplicative rate, not an additive rate. The base $\frac{3}{2}$ indicates that for each increase in $x$ by 1, the function value $f(x)$ is multiplied by $\frac{3}{2}$.

Step 3: Evaluate the Initial Value

The initial value of the function is the value of $f(x)$ when $x = 0$. Substituting $x = 0$ into the function:

\[ f(0) = 2\left(\frac{3}{2}\right)^{0} = 2 \times 1 = 2 \]

The initial value is 2, not 0.

Step 4: Analyze the Rate of Increase

The function does not increase by a constant additive amount. Instead, it increases by a constant multiplicative factor of $\frac{3}{2}$ as $x$ increases by 1.