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

Answer:

$g(x) = 3^{x}-6$

Explanation:

Step1: Analyze the growth rate of linear function

For the linear function $f(x)=2x + 10$, its growth rate is constant. The slope of the line is $m = 2$, so as $x$ increases, $f(x)$ increases by 2 for each unit increase in $x$.

Step2: Analyze the growth rate of exponential function

For the exponential function $g(x)=3^{x}-6$, the general form of an exponential function is $y = a^{x}$ ($a>1$). Here $a = 3$. The derivative of $y=a^{x}$ is $y^\prime=a^{x}\ln a$. As $x$ gets larger, the value of $3^{x}$ grows extremely rapidly. For example, when $x = 5$, $f(5)=2\times5+10=20$, and $g(5)=3^{5}-6=243 - 6=237$; when $x = 10$, $f(10)=2\times10 + 10=30$, and $g(10)=3^{10}-6=59049-6 = 59043$.

Since exponential functions ($a>1$) grow faster than linear functions as $x\rightarrow+\infty$, $g(x)=3^{x}-6$ will eventually exceed $f(x)=2x + 10$.