Inverse functions

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  • 1. Inverse FunctionsAP Calculus
  • 2. Definition Let f (x) have domain D and range R. The inverse function f -1 (x) (if it exists) is the function with domain R such that f -1 (f (x)) = x for x ϵ D f (f -1 (x)) = x for x ϵ R If f -1 exists then f is called invertible.
  • 3. Example: Linear Function Let f (x) = 4x – 1. Find f -1 (x) and show that f (x) is invertible. f -1 (x) = ¼ (x + 1) x f (x) x f -1 (x) 0 -1 -1 0 2 7 7 2 -2 -9 -9 -2 3 11 11 3 171 683 683 171
  • 4. Example: Linear Function Graphs of f and f -1
  • 5. Horizontal Line Testy = x2 is not one-to-one y = x3 is one-to-one
  • 6. Example: Function with noInverse y = x2 Is y = √x the inverse?? Is y = ±√x the inverse? x y = x2 x y = √x x y = ±√x -2 4 4 2 4 ±2 -1 1 1 1 1 ±1 0 0 0 0 0 0 1 1 1 1 1 ±1 2 4 4 2 4 ±2
  • 7. Example: Function with noinverse Graph of f and f -1
  • 8. One-to-one Function  Definition: A function f (x) is one-to- one (on its domain D) if for every value c, the equation f (x)=c has at most one solution for x ϵ D. a cDomain of f = Range of f -1 Range of f = Domain of f -1
  • 9. Theorem 1: Existence ofInverses If f (x) is one-to-one on its domain D then f is invertible. Domain of f = range of f -1 Range of f = domain of f -1
  • 10. Derivative of the Inverse Assume f (x) is invertible and one-to- one with the inverse g (x) = f -1 (x). If b belongs to the domain of g (x) and f ’ (g (b)) ≠ 0 then g ’ (b) exists and: