exponential function
An exponential function is a mathematical function of the form f(x) = a^x, where a is a positive constant called the base 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 called the base and x is the independent variable.
The exponential function is characterized by a rapid increase or decrease in the value of the function as the input variable changes. When a > 1, the function represents exponential growth, meaning that the output values increase as the input values increase. Conversely, when 0 < a < 1, the function represents exponential decay, meaning that the output values decrease as the input values increase. The base a determines the rate of growth or decay. If a > 1, the larger the value of a, the faster the function grows. On the other hand, if 0 < a < 1, the smaller the value of a, the faster the function decays. Exponential functions have several important properties: 1. Exponential form: As mentioned earlier, the function is in the form f(x) = a^x. The base, a, can be any positive number except 1, and x can be any real number. 2. Domain and range: The domain of an exponential function is all real numbers, (-∞, +∞). The range, however, depends on the base: if a > 1, the range is (0, +∞); if 0 < a < 1, the range is (0, +∞), but the function approaches 0 as x approaches infinity. 3. Graph: The graph of an exponential function depends on the value of the base. If a > 1, the graph will show exponential growth as x increases. The graph will be concave upward and will approach but never touch the x-axis. If 0 < a < 1, the graph will show exponential decay as x increases. The graph will be concave downward and will approach but never touch the x-axis. 4. Applications: Exponential functions are commonly used to model population growth, compound interest, radioactive decay, bacterial growth, and many other natural phenomena. To work with exponential functions, it is important to understand properties such as exponential equations, exponential growth/decay, and logarithms. These concepts will help you solve equations, graph functions, and analyze real-world problems involving exponential functions.
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