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. It represents growth or decay at a constant percentage rate.

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What does the base of an exponential function represent?

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The base of an exponential function (b in f(x) = a(b^x)) represents the growth factor. If b > 1, the function models exponential growth; if 0 < b < 1, it models exponential decay.

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How do you identify an exponential function from a table of values?

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An exponential function can be identified by checking if the ratio of consecutive outputs (y-values) is constant when the inputs (x-values) are increased by equal intervals.

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What is the general form of an exponential growth function?

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The general form of an exponential growth function is f(x) = a(b^x), where a > 0 and b > 1.

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What is the general form of an exponential decay function?

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The general form of an exponential decay function is f(x) = a(b^x), where a > 0 and 0 < b < 1.

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What is the y-intercept of the function f(x) = 2(3^x)?

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The y-intercept occurs when x = 0. Thus, f(0) = 2(3^0) = 2(1) = 2.

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What is the effect of changing the base in an exponential function?

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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 1) results in slower growth.

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How do you graph an exponential function?

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To graph an exponential function, plot the y-intercept, identify the growth or decay direction based on the base, and plot additional points by substituting x-values into the function.

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What is the domain and range of an exponential function?

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The domain of an exponential function is all real numbers (-∞, ∞). The range is (0, ∞) for growth functions and (0, a) for decay functions, where a is the initial value.

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What is the difference between linear and exponential functions?

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Linear functions have a constant rate of change (add/subtract the same amount), while exponential functions have a variable rate of change (multiply/divide by the same factor).