Exponential Function
f(x) = aˣ
An exponential function is a mathematical function of the form f(x) = a^x, where a must be a positive constant and x is any real number as its input. The base a is a positive number that determines the shape of the curve.
Exponential functions show rapid growth or decay, and they are often used to model real-life phenomena that exhibit exponential behavior. For example, the growth of bacteria in a Petri dish, the decay of radioactive materials, the growth of populations, the depreciation of assets, and the spread of diseases.
Properties of Exponential Functions:
1. Exponential functions are always positive. The output of the function is greater than zero for all positive x values.
2. The domain of the function is all real numbers.
3. The range of the function is all positive real numbers.
4. Exponential functions are increasing or decreasing depending on whether the base a is greater than or less than one, respectively.
5. The exponential function is one-to-one, meaning that each unique input corresponds to a unique output.
6. Exponential functions have a horizontal asymptote at y = 0, which means the function approaches but never reaches the x-axis.
Some common exponential functions:
1. f(x) = 2^x
2. g(x) = 10^x
3. h(x) = e^x (where e is Euler’s number, a mathematical constant approximately equal to 2.71828)
Exponential functions are often studied along with logarithmic functions, as they are inverse functions of each other. The logarithmic function is used to solve equations that involve exponential functions.
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