Definition of Exponential Function
An exponential function is a mathematical function of the form:
f(x) = a * bx,
where:
- f(x) represents the output of the function.
- x is the exponent or input variable.
- a is a constant, known as the initial value, which is not equal to zero.
- b is the base of the exponential function and is a positive number not equal to one. This base determines the growth or decay rate of the function.
In simpler terms, an exponential function grows (or decays) at a rate proportional to its current value, making it a vital model for various natural phenomena, such as population growth, radioactive decay, and interest calculations.
Characteristics of Exponential Functions
- When b > 1, the function is an exponential growth function, which means that as x increases, f(x) also increases rapidly.
- When 0 < b < 1, the function represents exponential decay, indicating that as x increases, f(x) decreases exponentially.
- The function has a horizontal asymptote at f(x) = 0, meaning it approaches but never touches the x-axis as x approaches negative infinity.
- The graph of an exponential function is always continuous and smooth, reflecting rapid changes in output value as input changes.
Overall, exponential functions are pivotal in many fields of study, including biology, finance, and physics, showcasing how values can grow or decline exponentially over time.