both of these functions grow as x gets larger and larger. which function eventually exceeds the other? f(x)…

Explanation:

Step1: Analyze function types

$f(x)=0.5(4)^{x}$ is an exponential - function with base $4>1$, and $g(x)=0.5x^{2}+6$ is a quadratic function.

Step2: Recall growth - rate property

Exponential functions with base $a > 1$ grow faster than polynomial functions as $x$ approaches infinity. That is, for an exponential function $y = a^{x}(a>1)$ and a polynomial function $y = b_{n}x^{n}+b_{n - 1}x^{n - 1}+\cdots + b_{1}x + b_{0}(n\geq0,b_{n}\neq0)$, $\lim_{x\rightarrow+\infty}\frac{a^{x}}{b_{n}x^{n}+b_{n - 1}x^{n - 1}+\cdots + b_{1}x + b_{0}}=+\infty$.

Answer:

$f(x)$ eventually exceeds $g(x)$.