1. DAY 7 – INVERSE OF A FUNCTION

2. 1.Use Exponential Regression to find an exponential function that contains the points (3, 54) and (4,162). 2.What is the initial value for this model? 3.What percentage growth or decay does this model imply ? WARM UP

3. KATHY AND KEVIN GRAPHED THE SAME DATA. BOTH INSIST THEY ARE CORRECT, BUT THEIR GRAPHS LOOK DIFFERENT. WHAT DO YOU THINK HAPPENED? LOOKING AT GRAPHS

4. Kathy and Kevin they switched their x and y values In Kathy’s graphIn Kevin’s Graph (0,1) (1,0) (2,4) (4,2) WHAT HAPPENED?

5. In mathematics, the inverse of a function occurs when the independent and dependent values of a function are reversed. We can create an inverse function by switching the x and y values. (6, 2) will become (2, 6), (-3, 1) becomes (1, -3). When we find an inverse function, we have to make sure it is still a function. INVERSE OF A FUNCTION

6. Remember, an inverse is an operation that take us back to the original input. A function is a mathematical relation where each input only has one corresponding output. Are each of these functions? Why or why not? DO ALL FUNCTIONS HAVE INVERSES?

7. For a function to have an inverse, each output must only have one corresponding input. Do these functions have inverses? Why or why not? DO ALL FUNCTIONS HAVE INVERSES?

8. How f(x) = y and g(y) = x compare? They have switched x and y. Since x and y have switch places, we say that f and g are inverses. A FUNCTION AND ITS INVERSE

9. If g is the inverse of f, we use the notation g = f -1 or g(x) = f -1 (x). The notation f -1 is read “f inverse of x.” IMPORTANT NOTATION

10. 1) f(x) = 6 + 3x2) f(x) = For each of the functions above, follow these steps  Make a table of 5 values and graph function 1 on graph paper.  Make a table of 5 values and graph function 2 the same graph.  What do you notice about the two tables?  What do you notice about the two graphs?  What line are the inverses reflected over? Write your conjecture on the line below. To graph the inverse of a function, you can reflect the original function over the line y = x OR make a table using the function, and then make a new table and switch the x and y coordinates. Now just graph the new table of points! PART 1: GRAPHS OF INVERSE FUNCTIONS

11. 1.Change f(x) notation to y notation 1.So f(x) = 3 + x becomes y = 3 + x 2.Switch the x and the y variables in the function 3.Solve the equation for y. 4.Replace y with FINDING THE INVERSE EQUATION

12. You can check your work by putting your original and inverse functions in the calculator. If they are reflected over y = x, you’ll know you’ve done it right! PART 2: EQUATIONS OF INVERSE FUNCTIONS

13. Complete the worksheet HOMEWORK