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

Explanation:

Step1: Analyze function types

Exponential functions grow faster than polynomial functions as (x) approaches infinity. (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.

Step2: Recall growth - rate rules

The general rule for the growth rate of functions as (x\to+\infty) is that exponential functions (y = a\cdot b^{x}(a\gt0,b > 1)) grow faster than polynomial functions (y=a_{n}x^{n}+a_{n - 1}x^{n - 1}+\cdots+a_{1}x + a_{0}(a_{n}\neq0,n\in N)). Here, for (f(x)=\frac{1}{2}(5)^{x}), (a=\frac{1}{2}), (b = 5>1); for (g(x)=5x^{2}), (n = 2), (a_{2}=5); for (h(x)=5x+\frac{9}{2}), (n = 1), (a_{1}=5).

Answer:

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