## Exponential Function (Graph)

### An exponential function is a mathematical function of the form f(x) = a^x, where a is a positive constant and x is the independent variable

An exponential function is a mathematical function of the form f(x) = a^x, where a is a positive constant and x is the independent variable. The graph of an exponential function is characterized by a rapid increase or decrease in value as x changes.

The graph of an exponential function can take two different shapes, depending on the value of a:

1. When a > 1: If the base, a, is greater than 1, the exponential function will exhibit exponential growth. As x increases, the function value will rapidly grow towards positive infinity. The graph will start from a point close to the x-axis and then steadily increase as x moves to the right.

Example: f(x) = 2^x, where a = 2. The graph of this exponential function will start at (0,1) and gradually rise as x increases.

2. When 0 < a < 1: If the base, a, is between 0 and 1, the exponential function will exhibit exponential decay. As x increases, the function value will rapidly decrease towards zero. The graph will start at a point close to the y-axis and then gradually decline as x moves to the right. Example: f(x) = (1/2)^x, where a = 1/2. The graph of this exponential function will start at (0,1) and gradually decline as x increases. Key features of exponential functions: 1. Domain: The domain of an exponential function is the set of all real numbers. 2. Range: The range of an exponential function is either all positive real numbers (if a > 1) or all positive numbers between 0 and 1 (if 0 < a < 1). 3. Vertical Asymptote: The graph of an exponential function will never touch or cross the x-axis. It may, however, approach the x-axis from either above or below, depending on the type of exponential growth or decay. 4. Horizontal Asymptote: When x approaches negative or positive infinity, the value of f(x) will approach 0. Therefore, the x-axis acts as a horizontal asymptote for all exponential functions. 5. Intercepts: The graph will intersect the y-axis at (0, 1) for all exponential functions. 6. Rate of Growth/Decay: The rate at which an exponential function grows or decays is determined by the value of the base, a. The larger a is, the more rapid the growth, while a value between 0 and 1 will lead to a more rapid decay. It is important to note that exponential functions are widely used in various fields like finance, biology, physics, and computer science, as they can model population growth, compound interest, radioactive decay, and more.

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