AP Precalculus Exponential Functions Review

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

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What is the base of an exponential function?

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The base of an exponential function is the constant 'b' in the function f(x) = a * b^x. It determines the rate of growth or decay of the function.

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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' is greater than 1, leading to an increase in the function's value as 'x' increases. Exponential decay occurs when the base 'b' is between 0 and 1, leading to a decrease in the function's value as 'x' increases.

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How do you identify the horizontal asymptote of an exponential function?

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The horizontal asymptote of an exponential function f(x) = a * b^x is typically y = 0, which means the function approaches but never reaches this line as x approaches negative infinity.

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What is the natural exponential function?

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The natural exponential function is an exponential function with base 'e' (approximately 2.71828), expressed as f(x) = e^x. It is widely used in mathematics, particularly in calculus.

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What is the inverse of an exponential function?

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The inverse of an exponential function is a logarithmic function. For example, if f(x) = a * b^x, then its inverse is f^(-1)(x) = log_b(x/a).

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How do you solve exponential equations?

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To solve exponential equations, you can use logarithms to isolate the variable. For example, if you have b^x = k, you can take the logarithm of both sides: x = log_b(k).

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What is the compound interest formula and how does it relate to exponential functions?

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The compound interest formula A = P(1 + r/n)^(nt) is an example of an exponential function, 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 continuous growth formula?

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The continuous growth formula is A = Pe^(rt), where A is the amount after time t, P is the initial amount, r is the growth rate, and e is the base of the natural logarithm.

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How do you graph an exponential function?

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To graph an exponential function, identify the y-intercept (when x=0), determine the horizontal asymptote, and plot points for various values of x to show the growth or decay behavior.