exponential functions
Exponential functions are a type of mathematical function that models the growth or decay of a quantity that increases or decreases at a constant rate over time
Exponential functions are a type of mathematical function that models the growth or decay of a quantity that increases or decreases at a constant rate over time. These functions have the general form:
f(x) = a * b^x
where “a” is the initial value or starting point, “b” is the base of the exponential function, and “x” represents the independent variable.
In exponential functions, the base “b” is a positive constant greater than 1. If the base is between 0 and 1, it represents exponential decay, while if it is greater than 1, it represents exponential growth.
The graph of an exponential function typically exhibits a characteristic curve, with either an upward trend (exponential growth) or a downward trend (exponential decay). The curve becomes steeper as the value of the base “b” increases.
Exponential functions have several important properties:
1. Domain and range: The domain of an exponential function is all real numbers, while the range depends on whether it represents growth or decay. If it represents growth, the range is all positive real numbers. If it represents decay, the range is all positive real numbers less than the initial value “a”.
2. Asymptote: Exponential functions have a horizontal asymptote at y = 0 (if it represents growth) or at y = a (if it represents decay). This means that the function approaches but never reaches this horizontal line.
3. Rate of change: Exponential functions have a constant rate of change that is proportional to the value of “x”. For growth, the rate of change is positive, while for decay, it is negative.
4. Compounding: Exponential functions are often used to model compound interest or exponential growth in various fields, such as finance, population growth, and radioactive decay. The base “b” is commonly referred to as the growth or decay factor.
To better understand exponential functions, it is helpful to examine examples and practice solving problems involving exponential growth and decay.
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