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, and polynomial functions of higher - degree grow faster than those of lower - degree. The function $f(x)=\frac{3}{2}x$ is a linear function (a polynomial of degree 1), $h(x)=\frac{1}{2}x^{2}$ is a quadratic function (a polynomial of degree 2), and $g(x)=2^{x}-4$ is an exponential function.

Step2: Compare growth rates

As $x$ approaches infinity, the exponential function $y = a^{x}$ ($a>1$) grows faster than any polynomial function $y = b_{n}x^{n}+b_{n - 1}x^{n - 1}+\cdots + b_{1}x + b_{0}$, where $n$ is a non - negative integer and $b_{i}$ are constants. Here, $a = 2$, $n = 2$ for $h(x)$ and $n = 1$ for $f(x)$.

Answer:

$g(x)=2^{x}-4$