Exponential Functions (Word Problems)

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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 processes.

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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 by a constant percentage over a period of time, resulting in a rapid increase. The function has a base greater than 1.

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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 by a constant percentage over a period of time, leading to a rapid decrease. The function has a base between 0 and 1.

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How do you write an exponential growth function?

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An exponential growth function can be written as f(t) = a(1 + r)^t, where 'a' is the initial amount, 'r' is the growth rate, and 't' is time.

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How do you write an exponential decay function?

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An exponential decay function can be written as f(t) = a(1 - r)^t, where 'a' is the initial amount, 'r' is the decay rate, and 't' is time.

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What is the formula for compound interest?

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The formula for compound interest is A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, P is the principal amount, r is the annual interest rate, n is the number of times that interest is compounded per year, and t is the number of years.

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If a population of bacteria triples every hour, how can we express this mathematically?

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The population can be expressed as P(t) = P_0(3^t), where P_0 is the initial population and t is the time in hours.

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What is the initial value in an exponential function?

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The initial value in an exponential function is the value of the function when the independent variable (usually time) is zero, represented by 'a' in the function f(x) = a(b^x).

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How do you determine the growth rate from an exponential function?

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The growth rate can be determined from the base of the exponential function. If the function is in the form f(x) = a(b^x), the growth rate is (b - 1) * 100%.

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

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The base of 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.