Question

find the equation of the exponential function.

x	f(x)
1	3·2
2	3·(2)²
3	3·(2)³

show your work here

hint: to add an exponent (xⁿ), type \exponent\ or press \^\

Explanation:

Step1: Recall exponential function form

The general form of an exponential function is \( f(x) = a \cdot b^x \), where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the base (the common ratio between consecutive terms).

Step2: Find the value of \( a \)

When \( x = 0 \), \( f(0) = 3 \). Substituting into the general form: \( f(0) = a \cdot b^0 = a \cdot 1 = a \). So, \( a = 3 \).

Step3: Find the value of \( b \)

Looking at the table, when \( x = 1 \), \( f(1) = 3 \cdot 2 \); when \( x = 2 \), \( f(2) = 3 \cdot (2)^2 \); when \( x = 3 \), \( f(3) = 3 \cdot (2)^3 \). The base \( b \) is the factor by which \( f(x) \) changes when \( x \) increases by 1. From \( x = 0 \) to \( x = 1 \), \( f(x) \) goes from 3 to \( 3 \cdot 2 \), so \( b = 2 \).

Step4: Write the exponential function

Substituting \( a = 3 \) and \( b = 2 \) into the general form \( f(x) = a \cdot b^x \), we get \( f(x) = 3 \cdot 2^x \).

Answer:

\( f(x) = 3 \cdot 2^x \)