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 variable. The base 'b' determines the growth or decay rate.

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What is the difference between exponential growth and exponential decay?

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Exponential growth occurs when the base 'b' in the function f(x) = a * b^x is greater than 1, leading to an increase in value as 'x' increases. Exponential decay occurs when 'b' is between 0 and 1, resulting in a decrease in value as 'x' increases.

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What is the range of an exponential growth function?

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The range of an exponential growth function f(x) = a * b^x (where b > 1) is (0, ∞), meaning it can take any positive real number.

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What is the range of an exponential decay function?

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The range of an exponential decay function f(x) = a * b^x (where 0 < b < 1) is also (0, ∞), meaning it can take any positive real number.

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

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The graph of an exponential growth function rises steeply as 'x' increases, starting from a positive value and approaching zero as 'x' approaches negative infinity.

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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 as 'x' increases, starting from a positive value and approaching zero as 'x' approaches positive infinity.

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What is the y-intercept of an exponential function f(x) = a * b^x?

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The y-intercept of the function is the value of f(0), which equals 'a', since f(0) = a * b^0 = a.

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How do you determine if a function is exponential growth or decay from its equation?

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If the base 'b' in the function f(x) = a * b^x is greater than 1, it is exponential growth. If 'b' is between 0 and 1, it is exponential decay.

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What is the formula for continuous exponential growth?

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The formula for continuous exponential growth is f(t) = Pe^(rt), where P is the initial amount, r is the growth rate, and t is time.

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What is the formula for continuous exponential decay?

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The formula for continuous exponential decay is f(t) = Pe^(-rt), where P is the initial amount, r is the decay rate, and t is time.