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

each of these functions grows as x gets larger and larger. which function eventually exceeds the others?\n$f(x)=\\left(\\frac{7}{2}\\right)^x$\n$g(x)=\\frac{7}{2}x^2$\n$h(x)=\\frac{7}{2}x$

each of these functions grows as x gets larger and larger. which function eventually exceeds the others?\n$f(x)=\\left(\\frac{7}{2}\\right)^x$\n$g(x)=\\frac{7}{2}x^2$\n$h(x)=\\frac{7}{2}x$

Answer

Explanation:

Step1: Recall growth - rate of functions

Exponential functions (a^x) ((a> 1)) grow faster than polynomial functions (b_nx^n + b_{n - 1}x^{n-1}+\cdots + b_0) as (x\to+\infty). The function (f(x)=\left(\frac{7}{2}\right)^x) is an exponential function with base (a = \frac{7}{2}>1). The function (g(x)=\frac{7}{2}x^2) is a polynomial function of degree (n = 2), and (h(x)=\frac{7}{2}x) is a polynomial function of degree (n = 1).

Step2: Compare the growth - rates

As (x) gets larger and larger, exponential functions with bases greater than 1 grow faster than polynomial functions. So, the function (f(x)=\left(\frac{7}{2}\right)^x) will eventually exceed the functions (g(x)=\frac{7}{2}x^2) and (h(x)=\frac{7}{2}x).

Answer:

(f(x)=\left(\frac{7}{2}\right)^x)