4)) Both of these functions grow as \( x \) gets larger and larger. Which function eventually exceeds the other? \[ f(x)=\frac{5}{9}(2)^{x} g(x)=\frac{7}{2} x \]

To determine which function eventually exceeds the other as \( x \) grows, let's analyze each function. The function \( f(x) = \frac{5}{9}(2)^x \) is an exponential function, which grows faster than any polynomial function, including \( g(x) = \frac{7}{2}x \). As \( x \) approaches infinity, \( f(x) \) will outpace \( g(x) \) significantly, proving that \( f(x) \) eventually exceeds \( g(x) \). A useful approach to find their intersection point—where one function starts exceeding the other—is to set them equal: \(\frac{5}{9}(2)^x = \frac{7}{2}x\). While this equation may not have a straightforward algebraic solution, it can be solved graphically or numerically, showing that beyond a certain point, specifically around \( x = 5 \), \( f(x) \) takes over and continues to grow rapidly.