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 of the form $a^x$ ($a>1$) grow faster than polynomial functions of the form $b_nx^n + b_{n - 1}x^{n - 1}+\cdots+b_0$ as $x\to+\infty$.

Step2: Identify function types

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 2 and $h(x)=\frac{7}{2}x$ is a polynomial function of degree 1.

Answer:

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