Exponential Function

Definition

An exponential function is a mathematical function of the form $$f(x) = a \cdot b^{x}$$ where 'a' is a constant, 'b' is the base of the exponential and is greater than zero, and 'x' is the exponent. Exponential functions are characterized by their rapid growth or decay, depending on the value of the base 'b'. These functions have unique properties such as a horizontal asymptote and specific behavior in their graphs, which distinguish them from linear or polynomial functions.

5 Must Know Facts For Your Next Test

1. Exponential functions always pass through the point (0, a), since any number raised to the power of zero is 1.
2. The graph of an exponential function increases rapidly for bases greater than 1 and approaches zero for bases between 0 and 1.
3. Exponential functions have a domain of all real numbers and a range that varies based on the value of 'a'.
4. The derivative of an exponential function maintains the same functional form, which is a unique characteristic among functions.
5. In real-world applications, exponential functions often model populations, radioactive decay, and compound interest due to their inherent growth or decay patterns.

Review Questions

• How do changes in the base 'b' of an exponential function affect its graph?
  • Changing the base 'b' of an exponential function significantly alters its growth or decay rate. If 'b' is greater than 1, the function exhibits exponential growth, resulting in a steep increase as 'x' increases. Conversely, if 'b' is between 0 and 1, the function displays exponential decay, leading to a rapid decrease towards zero as 'x' increases. Therefore, understanding how different values of 'b' influence the graph helps in predicting its behavior.
• Compare and contrast exponential functions with linear functions in terms of their growth rates.
  • Exponential functions grow much faster than linear functions as 'x' increases. While linear functions increase by a constant amount (the slope) with each step along the x-axis, exponential functions increase by a factor determined by their base. For example, in an exponential function with base 2, every time 'x' increases by 1, the function value doubles. This sharp increase leads to dramatic differences in long-term behavior between these two types of functions.
• Evaluate how real-world scenarios can be modeled using exponential functions and describe one specific application.
  • Exponential functions are used extensively to model real-world scenarios involving growth or decay processes. One specific application is in population growth modeling. When resources are unlimited, populations can grow exponentially as individuals reproduce at a constant rate. This means that the population size can double over regular intervals. Understanding this dynamic helps ecologists predict future population sizes and make informed decisions about conservation efforts and resource management.

Related terms

Base: In the context of exponential functions, the base refers to the constant 'b' that determines the rate of growth or decay.

Horizontal Asymptote: A horizontal line that a graph approaches but never touches, which is commonly found in the graphs of exponential functions as they approach zero.

Growth and Decay: These terms refer to the behavior of exponential functions, where growth occurs when the base 'b' is greater than 1, and decay occurs when 'b' is between 0 and 1.