Complete the sentences about the functions f(x)=6x+9 and g(x)=3(3)^x.

Question

For small values of x, the function with greater values is ___. For instance, ___ whereas ___ . As the value of x increases, however, ___ eventually has greater values.

Answer 1

To complete the sentences regarding the functions \( f(x) = 6x + 9 \) and \( g(x) = 3(3^x) \), we can analyze their behavior for small and large values of \( x \).

For small values of \( x \), the function with greater values is \( f(x) \). For instance, when \( x = 0 \), \( f(0) = 6(0) + 9 = 9 \) whereas \( g(0) = 3(3^0) = 3(1) = 3 \). As the value of \( x \) increases, however, \( g(x) \) eventually has greater values.

This unequal growth rate happens because \( g(x) \) grows exponentially while \( f(x) \) is a linear function.

Thus, the completed sentences would read:

For small values of \( x \), the function with greater values is \( f(x) \). For instance, when \( x = 0 \), \( f(0) = 9 \) whereas \( g(0) = 3 \). As the value of \( x \) increases, however, \( g(x) \) eventually has greater values.