Question

Complete the sentences about the functions f(x)=5x+2 and g(x)=(1)/(3)(3)^(x). For small values of x, the function with greater values is For instance, , while As the value of x increases, however, eventually has greater values.

Answer 1

For small values of x, the linear function f(x) = 5x + 2 has greater values, while for large values of x, the exponential function g(x) = (1/3)(3)^x eventually exceeds the values of f(x) due to its rapid growth.

Step-by-step explanation:

The student is inquiring about comparing the output values of two different functions, f(x) = 5x + 2 and g(x) = \(\frac{1}{3}(3)^x\), for small versus large values of x.

For small values of x, the function with greater values is f(x). For instance, when x is 0, f(0) = 2, while g(0) = \(\frac{1}{3}\).

As the value of x increases, however, g(x) eventually has greater values. This is because g(x) grows exponentially, and after a certain point, the exponential growth will surpass the linear growth of f(x).

Answer 2

For small values of x, f(x) = 5x + 2 has greater values, and as x increases, g(x) = (1/3)(3)^x eventually has greater values.

For small values of x, the function with greater values is:

f(x) = 5x + 2

For small values of x, the function g(x) = (1/3)(3)^x has a smaller value because the exponent is positive, and the value of 3 is greater than 1.

As a result, raising 1/3 to a positive power will result in a smaller value.

As the value of x increases, however, eventually g(x) = (1/3)(3)^x has greater values. This is because, for large positive values of x, the exponent will be much larger than 1, and the value of 3 will be much smaller than 1. Raising 1/3 to this larger exponent will result in a larger value for g(x).

In summary, for small values of x, f(x) = 5x + 2 has greater values, and as x increases, g(x) = (1/3)(3)^x eventually has greater values.