Question

both of these functions grow as x gets larger and larger. which function eventually exceeds the other?
$f(x) = 3x + 8$
$g(x) = 6^{x - 3} + 5$

Explanation:

Step1: Analyze the types of functions

$f(x) = 3x + 8$ is a linear function with a slope of $3$. $g(x)=6^{x - 3}+5$ is an exponential function with base $6>1$.

Step2: Recall the growth rates of linear and exponential functions

For linear functions of the form $y = mx + b$ ($m>0$), the growth rate is constant (equal to $m$). For exponential functions of the form $y = a\cdot b^{x}+c$ ($b > 1,a>0$), the growth rate increases as $x$ increases. In the long run (as $x$ becomes very large), exponential functions with $b>1$ grow much faster than linear functions.

Step3: Compare the two functions as $x$ gets large

As $x$ approaches infinity, the term $6^{x - 3}$ in $g(x)$ will grow extremely rapidly because the base $6>1$ and the exponent $x - 3$ increases without bound. The linear term $3x$ in $f(x)$ grows at a constant rate. So, as $x$ gets larger and larger, $g(x)$ will eventually exceed $f(x)$.

Answer:

$g(x)=6^{x - 3}+5$ eventually exceeds $f(x) = 3x + 8$ as $x$ gets larger.