Understanding Exponential Functions: Properties, Growth vs Decay, and Applications

Exponential Function

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

An exponential function is a mathematical function of the form f(x) = a^x, where “a” is a constant and “x” is a variable. The base “a” is typically a positive number greater than 1, but it can also be a fraction between 0 and 1.

The graph of an exponential function can take various forms depending on the value of “a”. If “a” is greater than 1, the function represents exponential growth, where the function value increases rapidly as x increases. On the other hand, if “a” is between 0 and 1, the function represents exponential decay, where the function value decreases rapidly as x increases.

Here are a few important properties of exponential functions:

1. Domain and Range: The exponential function is defined for all real numbers as its domain. The range of the exponential function depends on the value of “a”. If “a” is greater than 1, the range is all positive real numbers. If “0 < a < 1", the range is between 0 and positive infinity. 2. Growth vs Decay: As mentioned earlier, if "a" is greater than 1, the function represents exponential growth. This means that the function value increases very rapidly as "x" increases. In contrast, if "a" is between 0 and 1, the function represents exponential decay, where the function value decreases rapidly as "x" increases. 3. Asymptote: The exponential function never touches the x-axis. Rather, it approaches it as "x" goes to negative or positive infinity. If "a" is greater than 1, the asymptote is the x-axis. If "0 < a < 1", the asymptote is the y-axis. 4. One-to-One Function: The exponential function is a one-to-one function, which means that each x-value corresponds to a unique y-value, and vice versa. This property allows for the existence of the inverse function, called the logarithmic function. It's worth noting that logarithms are closely related to exponential functions. They act as the inverse of an exponential function and can be used to solve exponential equations or simplify exponential expressions. Exponential functions have various applications in real-life situations, such as population growth, compound interest, radioactive decay, and more. Understanding their properties and behaviors can help in analyzing and solving problems in these contexts.

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