6 Determining Whether Two Functions are Inverses Two functions are inverses if the meet the followingdefinition.
7 Determining Whether Two Functions are Inverses - Example Determine whether fand g are inverse functions
8 Horizontal Line Test (page 245) The Horizontal Line Test is used to determine whether a function would have an inverse over its natural domain.If a horizontal line is drawn anywhere through the graph of a function and the horizontal line does not intersect the graph in more that one point, then the function passes the horizontal line test.When a function passes the horizontal line test, the function referred to as one-to-one function. The function is also said to be invertible.
9 Horizontal Line Test (page 245) Functions not passing the horizontal line test must have theirdomains restricted in order to work with their inverses.
14 Increasing or Decreasing Functions Have Inverses (page 246) If the graph of a function f is always increasing or always decreasing over the domain of f, then the function f has an inverse over its entire natural domain.The derivative of a function (slopes of the tangent lines) determines whether a function is increasing or decreasing over an interval.So, the following theorem suggest that we can determine whether or not a function has an inverse over its entire domain (passes the horizontal line test).
15 Example 8 (page 247)for all x.So, even though we know that f has an inverse, we can notProduce a formula for it.
16 Restricting the Domain to Make Functions Invertible (page 247)
17 Differentiability Implies Continuity. Chapter 3 Review ItemDifferentiability Implies Continuity.BUTContinuity DOES NOT Imply Differentiability
18 Continuity and Differentiability of Inverse Functions (page 248) If a function is differentiable over an interval, then it iscontinuous over that interval.If a function is continuous over an interval, it is notnecessarily differentiable. ( Corner point, Point ofvertical tangency, or Point of discontinuity.