Question

consider the exponential function $f(x) = 2(3^x)$ and its graph.
the initial value of the function is
the base of the function is
the function shows exponential
(image of a graph with $y = f(x)$ plotted, grid from -8 to 8 on both axes, curve rising from near x-axis on left to (0,2) and up)

Explanation:

Step1: Find Initial Value

The initial value of an exponential function \( f(x) = a(b^x) \) is the value when \( x = 0 \). Substitute \( x = 0 \) into \( f(x)=2(3^x) \): \( f(0)=2(3^0)=2(1)=2 \).

Step2: Identify the Base

In the exponential function form \( f(x) = a(b^x) \), \( b \) is the base. For \( f(x)=2(3^x) \), the base \( b = 3 \).

Step3: Determine Growth/Decay

Since the base \( 3>1 \), the exponential function shows exponential growth.

Answer:

The initial value of the function is \( 2 \).
The base of the function is \( 3 \).
The function shows exponential growth.