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3.2 Inverse Functions and Logarithms

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

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3.2 Inverse Functions and Logarithms

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  1. 3.2 Inverse Functions and Logarithms 3.3 Derivatives of Logarithmic and Exponential functions

  2. One-to-one functions • Definition: A function f is called a one-to-onefunction if it never takes on the same value twice; that is f(x1) ≠ f(x2) whenever x1 ≠ x2. • 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) = x3 is one-to-one but f(x) = x2 is not.

  3. Inverse functions • Definition: Let fbe 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) = x3 is f -1(x)=x1/3 • Cancellation equations:

  4. 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 - x3

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

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

  7. 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) forf(x) = x3 + x + 1 Solution: By inspection f(0)=1, thus f -1(1) = 0 Then

  8. is called the natural logarithm function. Logarithmic Functions 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:

  9. 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:

  10. Derivatives of Logarithmic and Exponential functions Examples on the board.

  11. 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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