Question

Which statement BEST describes why the exponential function exceeds the linear function?

A) The exponential function has no negative y values.

B) As exponential functions will always exceed a linear function.

C) The exponential function has a lower initial value than the linear function.

D) The linear function has a higher initial value than the exponential function.

Answer 1

Exponential functions exceed linear functions because they grow at a rate proportional to their current value, accelerating as x increases, whereas linear functions grow at a constant rate.

Step-by-step explanation:

The statement that BEST describes why the exponential function exceeds the linear function is not explicitly listed in the provided options. The key characteristic of exponential functions is not that they lack negative y-values or have a specific initial value relative to linear functions, but rather that their rate of increase is proportional to their current value, leading to a rapid increase as the x-values grow larger. Hence, over a long enough interval, an exponential function will exceed a linear function because the growth rate of the exponential function accelerates, while the linear function increases at a constant rate.

For example, comparing the exponential function y = 2x and the linear function y = 2x, the linear function will have higher values initially (at small x-values), but as x increases, the exponential function will grow much faster and will eventually surpass the linear function in value.