Question

both of these functions grow as x gets larger and larger. which function eventually exceeds the other? f(x) = 4^{x - 3} g(x) = 8x

Explanation:

Step1: Analyze the types of functions

$f(x) = 4^{x - 3}$ is an exponential function with base $4>1$, and $g(x)=8x$ is a linear function.

Step2: Recall the growth rates of functions

Exponential functions with base $> 1$ grow faster than linear functions as $x$ becomes very large. That is, for large enough $x$, the value of the exponential function will surpass the value of the linear function. Since $f(x)$ is exponential and $g(x)$ is linear, as $x$ gets larger and larger, $f(x)$ will eventually exceed $g(x)$.

Answer:

$f(x) = 4^{x - 3}$ (or the function $f(x)$) will eventually exceed $g(x)$.