Problem 58 - College Algebra

Question

How can you tell whether an exponential model describes exponential growth or exponential decay?

Step-by-Step Solution

Answer

An exponential model \( f(x) = ab^x \) represents exponential growth if the base \( b > 1 \), and it represents exponential decay if \( 0 < b < 1 \).

Step 1: Determine the Base

First, identify the base \( b \) of the exponential function \( f(x) = ab^x \). The base \( b \) is the number that is raised to \( x \). For example, if you have an exponential function \( f(x) = 2^x \), the base \( b \) is 2.

Step 2: Identify the Growth/Decay

Once the base \( b \) is identified, compare it to 1. If \( b > 1 \), the function represents exponential growth. If \( 0 < b < 1 \), the function represents exponential decay. For instance, for the function \( f(x) = 2^x \), since \( b = 2 \) and \( 2 > 1 \), the function represents exponential growth.