1. 4.1 – ONE-TO-ONE FUNCTIONS; INVERSE FUNCTIONS Target Goals: 1.Obtain the graph of the inverse function 2.Determine the inverse of a function

2. WHAT IS A ONE-TO-ONE FUNCTION? A function f is one-to-one if for each x in the domain of f there is exactly one y in the range and NO y in the range is the image of more than one x in the domain. One-to-one function: each x in the domain has one and only one image in the range. No y in the range is the image of more than one x. Not a one-to-one function: C is the image of both 3 and 4. A function that either always increases or always decreases is one-to-one.

3. HOW CAN YOU TEST TO SEE IF A FUNCTION IS ONE-TO-ONE? 1. Horizontal Line Test 2. One-to-one functions pass both VERTICAL and HORIZONTAL LINE TESTS. 3. Check to see if a function is ALWAYS INCREASING or ALWAYS DECREASING. Ex 1) Is one-to-one? Ex 2) Is one-to-one?

4. HOW DO YOU FIND THE INVERSE OF A FUNCTION? 1. Check that the function is one-to-one. An inverse only exists if the function is one- to-one. 2. Switch x and y and then solve for y.

5. HOW DO YOU VERIFY INVERSES? Let f be a one-to-one function. The inverse of f, denoted by f -1, is a function such that f(f -1 (x)) = x = f -1 (f(x)).

6. GRAPHING ONE-TO-ONE FUNCTIONS AND THEIR INVERSES Domain of f = range of f -1 Range of f = domain of f -1 The graph of a function and its inverse are symmetric with respect to the line y = x.

7. LET’S TRY THIS!!! Find the inverse of where. Then check the result by showing that

8. PARTNER WORK/PRACTICE For the following problems, the given function is one-to-one. A. Find the inverse and check your answer. B. State the domain and range for both the function and its inverse. C. Graph the function, its inverse, and y = x on the same coordinate axes on your calculator to visually check.

9. HW: 4.1 BOOKWORK