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

Explanation:

Step1: Recall growth - rate of functions

Exponential functions grow faster than polynomial functions as (x\to+\infty). The function (f(x)=3^{x}) is an exponential function, (g(x) = 5x^{2}-8) is a quadratic (polynomial) function and (h(x)=8x) is a linear (polynomial) function.

Step2: Analyze the growth - rate behavior

For polynomial functions (y = a_nx^n+\cdots+a_1x + a_0) ((a_n\neq0)) and exponential functions (y = a^x) ((a>1)), as (x) gets very large, the exponential function (y = a^x) will out - grow the polynomial function. The degree of (g(x)=5x^{2}-8) is (n = 2) and the degree of (h(x)=8x) is (n = 1). But no matter the degree of the polynomial, the exponential function (f(x)=3^{x}) will eventually exceed them.

Answer:

(f(x)=3^{x})