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3.2 Inverse Functions and Logarithms 3.3 Derivatives of Logarithmic and Exponential functions

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One-to-one functions Definition: A function f is called a one-to-one function if it never takes on the same value twice; that is f(x 1 ) ≠ f(x 2 ) whenever x 1 ≠ x 2. Horizontal line test: A function f is one-to-one if and only if no horizontal line intersects its graph more than once. Examples: f(x) = x 3 is one-to-one but f(x) = x 2 is not.

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Inverse functions Definition: Let f be a one-to-one function with domain A and range B. Then the inverse function f -1 has domain B and range A and is defined by for any y in B. Note: f -1 (x) does not mean 1 / f(x). Example: The inverse of f(x) = x 3 is f -1 (x)=x 1/3 Cancellation equations:

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How to find the inverse function of a one-to-one function f Step 1: Write y=f(x) Step 2: Solve this equation for x in terms of y (if possible) Step 3: To express f -1 as a function of x, interchange x and y. The resulting equation is y = f -1 (x) Example: Find the inverse of f(x) = 5 - x 3

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Another example: Switch x and y : Inverse functions are reflections about y = x. Solve for x :

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At x = 2 : At x = 4 : Slopes are reciprocals. First consider an example: Derivative of inverse function

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Calculus of inverse functions Theorem: If f is a one-to-one continuous function defined on an interval then its inverse function f -1 is also continuous. Theorem: If f is a one-to-one differentiable function with inverse function f -1 and f ′ (f -1 (a)) ≠ 0, then the inverse function is differentiable and Example: Find (f -1 ) ′ (1) for f(x) = x 3 + x + 1 Solution: By inspection f(0)=1, thus f -1 (1) = 0 Then

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Consider where a>0 and a≠1 This is a one-to-one function, therefore it has an inverse. The inverse is called the logarithmic function with base a. Example: The most commonly used bases for logs are 10: and e : is called the natural logarithm function. Logarithmic Functions

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Properties of Logarithms Since logs and exponentiation are inverse functions, they “un-do” each other. Product rule: Quotient rule: Power rule: Change of base formula:

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Derivatives of Logarithmic and Exponential functions Examples on the board.

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Logarithmic Differentiation The calculation of derivatives of complicated functions involving products, quotients, or powers can often be simplified by taking logarithms. Step 1: Take natural logarithms of both sides of an equation y = f (x) and use the properties of logarithms to simplify. Step 2: Differentiate implicitly with respect to x Step 3: Solve the resulting equation for y ′ Examples on the board

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