Question

Values for the function g(x) are shown in the table.


Which statement proves that is an exponential function?

Responses

All values of g(x) are fractions.
All values of g ( x ) are fractions.

All values of g(x) are positive numbers.
All values of g ( x ) are positive numbers.

The function g(x) decreases by equal factors over equal intervals.
The function g ( x ) decreases by equal factors over equal intervals.

The function g(x) grows by equal differences over equal intervals.

Answer 1

To determine if the function \( g(x) \) is an exponential function, we need to identify the characteristics that define exponential functions. One key feature of exponential functions is that they grow (or decrease) by equal factors over equal intervals.

Given the options presented, the appropriate statement that would prove that \( g(x) \) is an exponential function is:

**The function \( g(x) \) decreases by equal factors over equal intervals.**

This statement indicates that the function exhibits the characteristic of exponential decay, where the value decreases by a constant multiplicative factor for each equal increment in \( x \). If it were increasing, it would still hold true that it grows by equal factors, which is the hallmark of exponential growth.

In contrast, the other options relate to the nature of the values (being fractions or positive) but do not directly address the defining behavior of an exponential function.