Mathematics in Cybersecurity

9Exponentials and Algorithms

9.1 Exponential Functions

Exponential functions exhibit a rate of change—the change per instant or unit of time—proportional to the function's current value. If that rate of change is positive, the situation is described as exponential growth, while a negative change is described as exponential decay.

These functions are characterized by a constant base that is raised to a variable power. When we say a “variable power,” we refer to a situation where the exponent is an expression involving a variable, such as x. We shall find these functions play a crucial role in modeling a wide range of phenomena in various fields, such as finance, biology, chemistry, and physics.

The general form of an exponential function is:

f (x) = a · b^x

where

• a, the constant coefficient of the exponential, also happens to be the y-intercept of the graph, or the initial value of the function.
• b is the base of the exponential, a non-negative real number not equal to 1, whose value determines the rate of growth or decay.
• x is the variable exponent in the general form.

9.1.1 Properties of Exponential Functions When Stated in General Form

1. Increasing or decreasing: Exponential functions either increase or decrease throughout their domain, depending on the value of the exponential base b:
   • If b > 1, the function is increasing (exponential growth).
   • If 0 < b < 1, the function is decreasing (exponential decay).
2. Asymptotic behavior: Exponential functions of the form f(x) = a * ...