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

Answer:

$g(x)$

Explanation:

Step1: Analyze growth - rate of polynomial

The function $f(x)=x^{2}+x + 8$ is a polynomial function of degree 2. As $x$ gets large, the leading - term $x^{2}$ dominates its growth, and the growth rate of a polynomial function $y = a_nx^n+\cdots+a_0$ ($a_n\neq0$) is determined by its degree $n$.

Step2: Analyze growth - rate of exponential

The function $g(x)=4^{x - 3}+9$ is an exponential function. The general form of an exponential function is $y = a\cdot b^{x}+c$ ($a\neq0,b > 0,b\neq1$). For $g(x)$, the base $b = 4>1$. Exponential functions with $b>1$ grow faster than any polynomial function as $x$ approaches infinity.

Step3: Conclusion

As $x$ gets larger and larger, the exponential function $g(x)=4^{x - 3}+9$ will eventually exceed the polynomial function $f(x)=x^{2}+x + 8$.