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 the base of an exponential function represent?

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The base of an exponential function determines the rate of growth or decay. If the base is greater than 1, the function represents exponential growth; if the base is between 0 and 1, it represents exponential decay.

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What is the initial amount in an exponential function?

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The initial amount, denoted as 'a' in the function f(x) = a(b^x), represents the value of the function when x = 0.

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How do you solve exponential equations?

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To solve exponential equations, you can use properties of exponents, take logarithms, or rewrite the equation in a form that allows for comparison of bases.

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What is the formula for exponential decay?

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The formula for exponential decay is y = 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 range of an exponential function?

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The range of an exponential function f(x) = a(b^x) is (0, ∞) if a > 0, and (-∞, 0) if a < 0.

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How do you find the y-intercept of an exponential function?

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The y-intercept of an exponential function f(x) = a(b^x) is found by evaluating f(0), which equals 'a'.

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What is the relationship between exponential functions and logarithms?

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Exponential functions and logarithms are inverse functions. If y = b^x, then x = log_b(y).

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What is the formula for exponential growth?

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The formula for exponential growth is y = 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 express a number in exponential form?

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To express a number in exponential form, identify the base and the exponent that represent the number, such as 8 = 2^3.