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October 28, 2009

October 28, 2009. 5.3 Inverse Functions. Objectives:. TSWBA…. to verify that one function is the inverse function of another function; to determine whether a function has an inverse; to find the derivative of an inverse function. End. of. Slide.

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October 28, 2009

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  1. October 28, 2009

  2. 5.3 Inverse Functions Objectives: TSWBA… • to verify that one function is the inverse function of another function; • to determine whether a function has an inverse; • to find the derivative of an inverse function.

  3. End of Slide A function g is the inverse function of the function f if f(g(x)) = x for each x in the domain of g and g(f(x)) = x for each x in the domain of f. Definition of Inverse Function Show that the functions are inverse functions of each other.

  4. End of Slide The graph of f contains the point (a, b) if and only if the graph of its inverse contains the point (b, a). Reflective Property of Inverse Functions

  5. End of Slide A function has an inverse function if and only if it is one-to-one. If f is strictly monotonic on its entire domain, then it is one-to-one and therefore has an inverse function. The Existence of an Inverse Function Which of the following functions has an inverse function?

  6. End of Slide 1. Determine whether the function given by y = f(x) has an inverse function. Guidelines for Finding an Inverse Function 2. Interchange x and y. 3. Solve for y as a function of x. 4. Verify that your answer is in fact the inverse. Find the inverse of the following function.

  7. End of Slide Let f be a function whose domain in an interval I. If f has an inverse function, then the following statements are true. Continuity and Differentiability of Inverse Functions • If f is continuous on its domain, then f -1 is continuous on its domain. 2. If f is increasing on its domain, then f -1 is increasing on its domain. 3. If f is decreasing on its domain, then f -1 is decreasing on its domain. 4 If f is differentiable at c, then f -1 is differentiable at f(c).

  8. End of Slide Let f be a function that is differentiable on an interval I. If f has an inverse function g, then g is differentiable at any x. Moreover, The Derivative of an Inverse Function Find the derivative of the inverse of the following function. • What is the value of f-1(x) when x=3? • What is the value of (f-1)’(x) when x = 3?

  9. Find (f-1)’(a) for the function f and the given real number x. (a) (b)

  10. AP Calculus Problem 18%

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