Question

both of these functions grow as x gets larger and larger. which function eventually exceeds the other?
f(x) = 6x

g(x) = 3(2)^x - 5

Explanation:

Step1: Analyze function types

\( f(x) = 6x \) is a linear function (degree 1 polynomial), and \( g(x)=3(2)^x - 5 \) is an exponential function (base \( 2>1 \)).

Step2: Recall growth rates

Exponential functions with base \( > 1 \) grow faster than linear functions as \( x \to \infty \). So, for large enough \( x \), the exponential function \( g(x) \) will exceed the linear function \( f(x) \).

Answer:

\( g(x) = 3(2)^x - 5 \) eventually exceeds \( f(x) = 6x \)