What is the term for a function that consistently increases without bound?

A function that consistently increases without bound is described as exhibiting exponential growth. In mathematical terms, exponential growth occurs when the rate of change of a quantity is proportional to the value of that quantity itself. This is typically represented by a function in the form of ( f(x) = a \cdot b^x ), where ( b > 1 ) and ( a ) is a positive constant. As ( x ) increases, the output values of the function grow rapidly, leading to the increase being unbounded as ( x ) approaches infinity.

In contrast, a linear function produces a constant rate of change, and while it can increase indefinitely, its growth is predictable and not as rapid as exponential growth. Exponential decay, on the other hand, describes a situation where a quantity decreases over time rather than increasing. A quadratic function, defined by a power of 2, can increase but also has a fixed rate of acceleration due to its parabolic shape. Therefore, the nature of exponential growth is specifically characterized by the consistent and unbounded increase of the function, making it the correct choice.