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

Explanation:

Step1: Analyze growth - rate of polynomial function

The function $f(x)=x^{2}+3x + 10$ is a polynomial function of degree 2. Polynomial functions of degree $n$ grow as $x^n$ for large $x$. So, $f(x)$ grows like $x^{2}$ for large $x$.

Step2: Analyze growth - rate of exponential function

The function $g(x)=3^{x}$ is an exponential function. Exponential functions $a^{x}$ ($a>1$) grow faster than any polynomial function as $x$ approaches infinity.

Step3: Compare growth - rates

As $x$ gets larger and larger, the exponential function $g(x)=3^{x}$ will eventually exceed the polynomial function $f(x)=x^{2}+3x + 10$.

Answer:

$g(x)=3^{x}$ eventually exceeds $f(x)=x^{2}+3x + 10$.