Question

both of these functions grow as x gets larger and larger. which function eventually exceeds the other?
$f(x) = 4(3)^x$
$g(x) = 20x$

Explanation:

Step1: Analyze the types of functions

$f(x) = 4(3)^x$ is an exponential function with base $3>1$, so it grows exponentially. $g(x)=20x$ is a linear function with a slope of $20$.

Step2: Recall the growth rates

Exponential functions with base $> 1$ grow faster than linear functions as $x$ becomes large. For example, when $x = 5$:

• $f(5)=4(3)^5=4\times243 = 972$
• $g(5)=20\times5 = 100$

When $x = 10$:

• $f(10)=4(3)^{10}=4\times59049 = 236196$
• $g(10)=20\times10 = 200$

As $x$ increases, the exponential function $f(x)$ will eventually exceed the linear function $g(x)$.

Answer:

$f(x) = 4(3)^x$