Question

both of these functions grow as x gets larger and larger. which function eventually exceeds the other?
f(x) = 4^{x - 2}
g(x) = 5x + 8

Explanation:

Step1: Identify function types

$f(x)=4^{x-2}$ is exponential; $g(x)=5x+8$ is linear.

Step2: Compare growth rates

Exponential functions grow faster than linear functions as $x\to\infty$. To verify, test large $x$:
For $x=5$: $f(5)=4^{5-2}=4^3=64$, $g(5)=5(5)+8=33$
For $x=6$: $f(6)=4^{6-2}=4^4=256$, $g(6)=5(6)+8=38$

Step3: Conclude long-term behavior

As $x$ increases, $f(x)$ outpaces $g(x)$.

Answer:

The function $f(x)=4^{x-2}$ eventually exceeds $g(x)=5x+8$.