Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Functions
Exponential functions are mathematical expressions in the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. The function f(x) = e^x is a specific case where the base 'b' is Euler's number (approximately 2.718). These functions exhibit rapid growth or decay, depending on the value of 'b', and are characterized by their continuous and smooth curves.
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Transformations of Functions
Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. For example, the function h(x) = e^-x represents a reflection of f(x) = e^x across the y-axis. Understanding these transformations allows one to manipulate the original graph to obtain the graph of a new function while maintaining the same general shape.
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Asymptotes
Asymptotes are lines that a graph approaches but never touches or crosses. For exponential functions, the horizontal asymptote is typically found at y = 0, indicating that as x approaches negative infinity, the function's value approaches zero. Identifying asymptotes is crucial for understanding the behavior of the graph, particularly in determining the domain and range of the function.
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