Both of these functions grow as x gets larger and larger. Which function eventually exceeds the other? f(x)=(5)/(7)(4)^x g(x)=4x+(20)/(7)

To determine which function eventually exceeds the other as gets larger, we need to analyze the growth rates of the two functions.First, let's examine the function \( f(x) = \frac{5}{7}(4)^x \). This is an exponential function with a base of 4. Exponential functions grow much faster than linear functions as increases.Next, let's look at the function \( g(x) = 4x + \frac{20}{7} \). This is a linear function with a slope of 4. Linear functions grow at a constant rate, which is slower compared to the exponential growth of \( f(x) \).To provide a more concrete comparison, let's calculate the values of both functions for a few larger values of :For : For : As we can see, even for relatively small values of , \( f(x) \) grows significantly faster than \( g(x) \). This trend will continue as becomes larger.Therefore, the function \( f(x) = \frac{5}{7}(4)^x \) will eventually exceed the function \( g(x) = 4x + \frac{20}{7} \) as gets larger.