each of these functions grows as x gets larger and larger. which function eventually exceeds the others…

Explanation:

Step1: Analyze the function types

$f(x)=8x + 6$ is a linear - function of the form $y=mx + b$ where $m = 8$ and $b = 6$. As $x$ increases, its growth rate is constant (the slope $m$).

Step2: Analyze the second function

$g(x)=4^{x - 3}=\frac{4^{x}}{4^{3}}=\frac{1}{64}\times4^{x}$ is an exponential function of the form $y = a\times b^{x}$ with $a=\frac{1}{64}$ and $b = 4$. Exponential functions with $b>1$ grow faster than polynomial functions as $x$ gets large.

Step3: Analyze the third function

$h(x)=3x^{2}-4x - 6$ is a quadratic polynomial function of the form $y = ax^{2}+bx + c$ with $a = 3$, $b=-4$ and $c=-6$. Polynomial functions have a fixed - degree growth rate.

Step4: Compare growth rates

As $x$ approaches infinity, exponential functions grow faster than polynomial functions. Among linear ($f(x)$), quadratic ($h(x)$) and exponential ($g(x)$) functions, the exponential function $g(x)=4^{x - 3}$ will eventually exceed the others.

Answer:

$g(x)=4^{x - 3}$