Exponential Function

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.

Key Concepts:

1. Base (b): The base of an exponential function is the constant raised to the power of x.
2. Exponent (x): The exponent represents the power to which the base is raised.
3. Growth and Decay: Exponential functions can model both growth (when b > 1) and decay (when 0 < b < 1).

Graphing Exponential Functions:

To graph an exponential function, you can plot points or use transformations of the basic exponential function f(x) = b^x. When b > 1, the graph increases from left to right. When 0 < b < 1, the graph decreases from left to right.

Properties of Exponential Functions:

1. Domain: The domain of an exponential function is all real numbers.
2. Range: The range of an exponential function depends on the value of b. When b > 1, the range is all positive real numbers. When 0 < b < 1, the range is all positive real numbers less than 1.
3. Asymptote: The x-axis (y = 0) is a horizontal asymptote for the graph of an exponential function.

Common Exponential Functions:

Some common exponential functions include:

• Exponential growth: f(x) = a * b^x, where b > 1
• Exponential decay: f(x) = a * b^x, where 0 < b < 1

Real-World Applications:

Exponential functions can be used to model population growth, radioactive decay, compound interest, and more.

Study Guide:

1. Understand the basic form of an exponential function: f(x) = a * b^x
2. Identify the base and exponent in an exponential function
3. Graph exponential functions and understand their behavior based on the value of the base
4. Explore real-world applications of exponential functions
5. Practice solving problems involving exponential growth and decay

Exponential functions are an important topic in mathematics and have wide-ranging applications in various fields. Understanding the properties and behaviors of exponential functions can help in analyzing and solving real-world problems.