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 the base (a positive real number), and 'x' is the exponent.

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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 0 < b < 1.

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What is the common ratio in an exponential sequence?

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The common ratio in an exponential sequence is the factor by which each term is multiplied to get the next term.

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How can you identify exponential growth from its function?

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Exponential growth occurs when the base 'b' in the function f(x) = a * b^x is greater than 1 (b > 1).

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How can you identify exponential decay from its function?

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Exponential decay occurs when the base 'b' in the function f(x) = a * b^x is between 0 and 1 (0 < b < 1).

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What does the graph of an exponential decay function look like?

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The graph of an exponential decay function decreases rapidly at first and then levels off, approaching the x-axis but never touching it.

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What is the significance of the base in an exponential function?

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The base determines the rate of growth or decay; a larger base results in faster growth, while a base between 0 and 1 results in decay.

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What is the effect of a negative exponent in an exponential function?

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A negative exponent indicates a reciprocal; for example, b^(-x) = 1/(b^x), which can represent decay.

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What is the difference between exponential growth and linear growth?

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Exponential growth increases at a rate proportional to its current value, while linear growth increases by a constant amount.

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What is a real-world example of exponential decay?

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A real-world example of exponential decay is radioactive decay, where the quantity of a radioactive substance decreases over time.