both of these functions grow as x gets larger and larger. which function eventually exceeds the other? f(x)…

Explanation:

Step1: Analyze growth - rate of polynomial and exponential functions

Polynomial functions like (f(x)=6.5x^{2}+11) have a power - based growth rate. Exponential functions like (g(x)=6^{x}+7) have an exponential growth rate.

Step2: Recall the growth - rate rule

As (x) approaches infinity, exponential functions (y = a^{x}) ((a>1)) grow faster than polynomial functions (y = b_{n}x^{n}+b_{n - 1}x^{n - 1}+\cdots + b_{1}x + b_{0}), where (n) is a non - negative integer and (b_{i}) are constants. In (f(x)=6.5x^{2}+11), it is a polynomial function of degree (n = 2). In (g(x)=6^{x}+7), it is an exponential function with base (a = 6>1).

Answer:

The function (g(x)=6^{x}+7) eventually exceeds (f(x)=6.5x^{2}+11).