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

Explanation:

Step1: Analyze the growth rate of polynomial function

For the polynomial function (f(x)=x^{2}+4), the degree is (2). Polynomial functions grow based on their degree. As (x) increases, the term (x^{2}) dominates. But polynomial growth (even for high - degree polynomials) is slower compared to exponential growth.

Step2: Analyze the growth rate of exponential function

For the exponential function (g(x)=2^{x}-4), the base (a = 2>1). The general form of an exponential function is (y = a^{x}+c) ((a>1)). As (x\to+\infty), the term (2^{x}) dominates. The growth rate of an exponential function (y=a^{x}) ((a > 1)) is much faster than any polynomial function (y = b_{n}x^{n}+b_{n - 1}x^{n-1}+\cdots+b_{1}x + b_{0}), (n\in N).

We can also check by taking some large values of (x). Let (x = 10): (f(10)=10^{2}+4=100 + 4=104) (g(10)=2^{10}-4=1024-4 = 1020) Let (x = 20): (f(20)=20^{2}+4=400+4 = 404) (g(20)=2^{20}-4=1048576-4=1048572)

Answer:

(g(x)=2^{x}-4) eventually exceeds (f(x)=x^{2}+4)