both of these functions grow as x gets larger and larger. which function eventually exceeds the…

Answer:

The function $g(x)=6^{x - 3}+5$ eventually exceeds $f(x)=3x + 8$.

Explanation:

Step1: Identify function types

$f(x)$ is a linear function, $f(x)=3x + 8$ with a constant - rate of change.

Step2: Identify function types

$g(x)$ is an exponential function, $g(x)=6^{x - 3}+5$. The general form of an exponential function is $y = a\cdot b^{x}+c$ ($a\neq0$, $b>0,b\neq1$). Here $a = 1$, $b = 6$, $c = 5$.

Step3: Analyze growth rates

Exponential functions with $b>1$ grow faster than linear functions as $x$ approaches infinity. As $x$ gets larger and larger, the value of $6^{x - 3}$ in $g(x)$ will increase at a much faster rate than the linear growth of $3x$ in $f(x)$. So, $g(x)$ will eventually exceed $f(x)$.