Question

consider the exponential function f(x) = 2(3^x) and its graph. the initial value of the function is dropdown. the base of the function is dropdown with options 2, 3, 6, 8. the function shows exponential dropdown. (graph of y = f(x) is a curve starting near x-axis, rising through (0,2) approximately, with grid axes.)

Explanation:

Step1: Find Initial Value

The general form of an exponential function is \( f(x) = a(b^x) \), where \( a \) is the initial value (when \( x = 0 \)). For \( f(x)=2(3^x) \), when \( x = 0 \), \( f(0)=2(3^0)=2(1)=2 \). So initial value is 2.

Step2: Find Base of Function

In \( f(x)=2(3^x) \), the base of the exponential part \( b^x \) is 3. So base is 3.

Step3: Determine Exponential Behavior

Since the base \( 3>1 \), the 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.