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

Explanation:

Step1: Analyze function types

$f(x)=10x - 3$ is a linear - function of the form $y = mx + b$ where $m = 10$ and $b=-3$.

Step2: Analyze second function

$g(x)=4^{x - 3}+7$ is an exponential function of the form $y = a\cdot b^{x - h}+k$, here $a = 1$, $b = 4$, $h = 3$, $k = 7$.

Step3: Analyze third function

$h(x)=x^{2}+3x + 6$ is a quadratic function of the form $y=ax^{2}+bx + c$ with $a = 1$, $b = 3$, $c = 6$.

Step4: Recall growth - rate of functions

Exponential functions ($y = a\cdot b^{x}$, $b>1$) grow faster than polynomial functions (like linear $y=mx + b$ and quadratic $y = ax^{2}+bx + c$) as $x$ approaches infinity.

Answer:

$g(x)=4^{x - 3}+7$