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

Answer:

$f(x)=\frac{1}{2}(5)^x$

Explanation:

Step1: Recall growth - rate of functions

Exponential functions grow faster than polynomial functions as $x$ approaches infinity.

Step2: Identify function types

$f(x)=\frac{1}{2}(5)^x$ is an exponential function, $g(x) = 5x^2$ is a quadratic (polynomial) function, and $h(x)=5x+\frac{9}{2}$ is a linear (polynomial) function.

Step3: Compare growth - rates

As $x$ gets larger and larger, the exponential function $f(x)=\frac{1}{2}(5)^x$ will eventually exceed the polynomial functions $g(x)$ and $h(x)$.