Solve for as a function of = = x y: x g(y) f Interchange and The resulting equation is = x y. y f Define the domain of to be the of f f. Verify that = and = f(f x f x. Find the inverse (if it exists) of Example 3: (x) = 4x 5 . f = 0.25x + 1.25 f Let be a function that is f Derivative of an Inverse Function (Pages differentiable on an interval Let be a function whose domain is an interval If has an f I. f If has an inverse function I. f inverse function, then the following statements are true. then is g, g If is continuous on its domain, then is continuous on its f f domain. differentiable at any for x If is increasing on its domain, then is increasing on its f f which 0 f domain. If is decreasing on its domain, then is decreasing on its f f domain. . Moreover, If is differentiable on an interval containing and f c f 0, then is differentiable at f f(c). What you should learn How to find the derivative of an inverse function f = 1 , 0 . f This last theorem can be interpreted to mean that graphs of inverse functions have reciprocal slopes .
Solve for x as a function of y: x = g(y) = f Interchange x and y. The resulting equation is y = f Define the domain of f to be the of f. Verify that f(f = x and f = x. Example 3: Find the inverse (if it exists) of f = 0.25x + 1.25 f (x) = 4x 5 . Derivative of an Inverse Function (Pages Let f be a function whose domain is an interval I. If f has an inverse function, then the following statements are true. If f is continuous on its domain, then f is continuous on its domain. If f is increasing on its domain, then f is increasing on its domain. If f is decreasing on its domain, then f is decreasing on its domain. If f is differentiable on an interval containing c and f 0, then f is differentiable at f(c). 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 for which f 0 . Moreover, What you should learn How to find the derivative of an inverse function = 1 , f f 0 . This last theorem can be interpreted to mean that graphs of inverse functions have reciprocal slopes .
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