OBJECTIVES:  Find inverse functions and verify that two functions are inverse functions of each other.  Use graphs of functions to determine whether functions have inverse functions.  Use the Horizontal Line Test to determine whether functions are one-to-one.

  Let be two functions such that:  Under these conditions, the function g is the inverse function of the function. The function is denoted by. So,  The domain of must be equal to the range of, and the range of must be equal to the domain of Definition of Inverse Function

  Which of the functions is the inverse function of  a) b) EX 1: Verifying Inverse Functions

  The graphs of a function and its inverse function are related to each other in the following way:  If the point lies on the graph of, then the point must lie on the graph of, and vice versa.  The graph of is a reflection of the graph of in the line.  Horizontal Line Test for Inverse Functions  A function has an inverse function if and only if no horizontal line intersects the graph of at more than one point. Graph of an Inverse Function

  a)  b)  c) EX 2: Use the graph of f to determine whether the function has an inverse function

  A function is one-to-one when each value of the dependent variable corresponds to exactly one value of the independent variable. A function has an inverse function if and only if is one-to-one.  If the graph passes the Vertical Line Test and the Horizontal Line Test, then the function is one-to-one. One–to–One Functions

  1.Use the Horizontal Line Test to decide whether has an inverse function.  2.In the equation for, replace with.  3.Interchange the roles of, and solve for.  4.Replace with in the new equation. Finding an Inverse Function

  a)b) EX 3: Find the inverse function