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 does it mean for a function to exhibit exponential growth?

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Exponential growth occurs when a quantity increases at a rate proportional to its current value, typically represented by a function of the form f(x) = a(1 + r)^x, where r is the growth rate.

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

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Exponential decay occurs when a quantity decreases at a rate proportional to its current value, typically represented by a function of the form f(x) = a(1 - r)^x, where r is the decay rate.

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How do you determine the rate of depreciation in an exponential decay function?

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The rate of depreciation can be determined by the formula y = a(1 - r)^x, where 'r' is the rate of depreciation expressed as a decimal.

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What is the general form of an exponential decay equation?

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The general form of an exponential decay equation is y = a(1 - r)^x, where 'a' is the initial amount and 'r' is the decay rate.

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What is the general form of an exponential growth equation?

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The general form of an exponential growth equation is y = a(1 + r)^x, where 'a' is the initial amount and 'r' is the growth rate.

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How do you calculate the future value of an investment using an exponential growth function?

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To calculate the future value of an investment, use the formula A = P(1 + r)^t, where A is the amount of money accumulated after n years, P is the principal amount, r is the annual interest rate, and t is the time in years.

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

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The base in an exponential function determines the rate of growth or decay; a base greater than 1 indicates growth, while a base between 0 and 1 indicates decay.

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What is the range of the function y = 4^(x + 1) - 5?

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The range of the function y = 4^(x + 1) - 5 is (-5, ∞), meaning the output values can be any number greater than -5.

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How do you identify whether a function is exhibiting growth or decay?

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To identify whether a function is exhibiting growth or decay, examine the base of the exponential function: if the base is greater than 1, it is growth; if the base is between 0 and 1, it is decay.