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. These functions exhibit rapid growth or decay, depending on the base. For example, f(x) = 3^x grows quickly as x increases, while g(x) = 3^(-x) represents exponential decay.
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Horizontal Asymptotes
A horizontal asymptote is a horizontal line that a graph approaches as x approaches positive or negative infinity. In the context of exponential functions, if the function approaches a constant value as x goes to infinity or negative infinity, that constant is the horizontal asymptote. In the given graph, the horizontal asymptote is y = -1, indicating that the function approaches this line but never actually reaches it.
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Determining Horizontal Asymptotes
Transformations of Functions
Transformations of functions involve shifting, reflecting, stretching, or compressing the graph of a function. For example, the function g(x) = 3^(x-1) represents a horizontal shift of the base function f(x) = 3^x to the right by 1 unit. Similarly, h(x) = 3^x - 1 indicates a vertical shift downward by 1 unit. Understanding these transformations is crucial for identifying the correct function based on its graph.
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