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Exploring Exponential Functions with e: Graphs and Properties

Dive into the graph and properties of exponential functions with the constant e, an irrational number similar to π. Understand how y = e^x behaves and its domain and range. Discover why e = 2.7182818... is a unique mathematical constant.

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Exploring Exponential Functions with e: Graphs and Properties

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  1. 1 n As n gets very large, interest is continuouslycompounded. Examine the graph of f(n)= (1 + )n. The function has a horizontal asymptote. As n becomes infinitely large, the value of the function approaches approximately 2.7182818…. This number is called e. Like , the constant e is an irrational number.

  2. Exponential functions with e as a base have the same properties as the functions you have studied. The graph of f(x) = ex is like other graphs of exponential functions, such as f(x) = 3x. The domain of f(x) = ex is all real numbers. The range is {y|y > 0}.

  3. Caution The decimal value of e looks like it repeats: 2.718281828… The value is actually 2.71828182890… There is no repeating portion.

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