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 \( x \) gets larger, we need to analyze the growth rates of the two functions.<br /><br />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 \( x \) increases.<br /><br />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 \( f(x) \).<br /><br />To provide a more concrete comparison, let's calculate the values of both functions for a few larger values of \( x \):<br /><br />For \( x = 10 \):<br />\[ f(10) = \frac{5}{7}(4)^{10} \approx \frac{5}{7} \times 1048576 \approx 7464960 \]<br />\[ g(10) = 4 \times 10 + \frac{20}{7} = 40 + \frac{20}{7} \approx 40 + 2.86 \approx 42.86 \]<br /><br />For \( x = 20 \):<br />\[ f(20) = \frac{5}{7}(4)^{20} \approx \frac{5}{7} \times 16777216 \approx 11957422.86 \]<br />\[ g(20) = 4 \times 20 + \frac{20}{7} = 80 + \frac{20}{7} \approx 80 + 2.86 \approx 82.86 \]<br /><br />As we can see, even for relatively small values of \( x \), \( f(x) \) grows significantly faster than \( g(x) \). This trend will continue as \( x \) becomes larger.<br /><br />Therefore, the function \( f(x) = \frac{5}{7}(4)^x \) will eventually exceed the function \( g(x) = 4x + \frac{20}{7} \) as \( x \) gets larger.