Exponential Functions

An exponential function is a mathematical function of the form f(x) = a * b^x, where a and b are constants and b is a positive real number not equal to 1. The base, b, is raised to the power of x, making the function grow at an increasing rate as x increases.

Key Concepts

1. Base: The base, b, determines the growth or decay rate of the function. If b > 1, the function shows exponential growth. If 0 < b < 1, the function shows exponential decay.
2. Exponential Growth: When the base, b, is greater than 1, the function increases at an increasing rate as x increases.
3. Exponential Decay: When the base, 0 < b < 1, the function decreases at an increasing rate as x increases.
4. Domain and Range: The domain of an exponential function is all real numbers, and the range depends on whether the function represents growth or decay.
5. Graphing Exponential Functions: Exponential functions have a characteristic curve that increases or decreases rapidly based on the value of the base, b.

Study Guide

To master the topic of exponential functions, it's important to focus on the following key areas:

1. Understanding the concept of exponential growth and decay.
2. Identifying the base of an exponential function and determining whether it represents growth or decay.
3. Exploring the domain and range of exponential functions.
4. Graphing exponential functions and recognizing their characteristic curves.
5. Solving problems involving exponential growth and decay.

Practice working with different values of a and b to understand their impact on the behavior of exponential functions.

Remember to seek help from your teacher or tutor if you encounter any challenges in understanding these concepts.