What characteristic defines exponential functions?

Exponential functions are characterized by the nature of their growth relative to the independent variable. The key feature is that they increase rapidly as the input value increases. This rapid increase occurs because the output of an exponential function is defined as a constant raised to the power of the variable input, leading to a situation where small increments in the input can result in significantly larger increases in output.

For example, in the function ( f(x) = a \cdot b^x ) (where ( a ) is a constant and ( b > 1 )), even a small increase in ( x ) leads to a substantial increase in ( f(x) ). This property is what sets exponential functions apart from linear or polynomial functions, which do not exhibit the same level of rapid increase as the variable grows.

The other options presented highlight various misunderstandings about exponential functions. A constant rate of change is a trait of linear functions. Referring to a variable raised to a non-negative integer describes polynomial functions more than exponentials. While exponential functions can produce negative outputs depending on the base and the exponent, it is not accurate to state that they can only have positive outputs, as they can also cross the x-axis when the parameters are chosen appropriately (