Characteristics of Exponential Functions

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What is an exponential function?

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An exponential function is a mathematical function of the form f(x) = a * b^x, where 'a' is a constant, 'b' is a positive real number, and 'x' is the exponent.

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What is the horizontal asymptote of an exponential function?

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The horizontal asymptote of an exponential function is a horizontal line that the graph approaches as x approaches positive or negative infinity. For functions of the form f(x) = a * b^x, the horizontal asymptote is typically y = 0.

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What is the range of an exponential function?

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The range of an exponential function f(x) = a * b^x (where a > 0) is (0, ∞) if the function approaches y = 0 as x approaches negative infinity.

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What is the y-intercept of an exponential function?

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The y-intercept of an exponential function f(x) = a * b^x is the point where the graph crosses the y-axis, which occurs at (0, a).

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What does it mean for an exponential function to exhibit growth?

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An exponential function exhibits growth when the base 'b' is greater than 1, resulting in the function increasing rapidly as x increases.

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What does it mean for an exponential function to exhibit decay?

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An exponential function exhibits decay when the base 'b' is between 0 and 1, resulting in the function decreasing rapidly as x increases.

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What is the domain of an exponential function?

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The domain of an exponential function is all real numbers, represented as (-∞, ∞).

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How does changing the base of an exponential function affect its graph?

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Changing the base of an exponential function affects the rate of growth or decay. A larger base results in faster growth, while a smaller base (greater than 0 but less than 1) results in slower growth or decay.

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What is the significance of the constant 'a' in an exponential function?

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The constant 'a' in an exponential function f(x) = a * b^x determines the initial value of the function when x = 0, which is also the y-intercept.

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What is an example of an exponential growth function?

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An example of an exponential growth function is f(x) = 2 * 3^x, where the base 3 indicates rapid growth.