1. 1.8 Inverse functions My domain is your range No! My range is your domain

2. Input/Output table to show how this works If g(x) = f -1 (x), then the range of f(x) is the domain of g(x). x f(x)x g(x) 1991 2882 3773 Can you think of the equations f(x) and g(x)?

3. f(x) = - x + 10 g(x) = - x + 10 To check to see if they are inverses we should use composition of functions. (f º g)(x) = - (- x + 10 ) + 10 = x – 10 + 10 so (f º g)(x) = x (g º f)(x) = - (- x + 10 ) + 10 Same problem so (g º f)(x) = x

4. Lets try another set of inverse functions f(x) = 3x + 2g(x) = 1/3x – 2/3 (f º g)(x) = 3(1/3x – 2/3) + 2 (f º g)(x) = x (g º f)(x) = 1/3(3x + 2) – 2/3 (g º f)(x) = x What can you say about inverse functions?

5. To check if 2 functions are inverse Use the compositions of the two functions, namely f(x) and g(x). (f º g)(x) = x (g º f)(x) = x The value will always be x

6. Graphing inverse functions h(x) = x 2 and f(x) = √x

7. The inverses reflects over the line y = x (Identity function) h(x) = x 2 and f(x) = √x This graph does not look complete why?

8. Horizontal Line Test If you pass a horizontal line through the graph and it only touches in one point, then the function has an inverse. Does not have an inverse, unless we limit the domain

9. Horizontal Line Test Does this function have an inverse?

10. One to One functions For every dependent variable (output) there is only one independent variable (input).Look at f(x) = 2x -3 If f(x) = 1, then x = 2 If x = 2, then f(x) = 1 A function only has an inverse if and only if it is “one to one”.

11. Would f(x) = x 2 have an inverse? If f(x) = 16, then x = 4 or – 4. Which makes f(x) not one to one, so no inverse.

12. How to find the Inverse Given f(x) = 8x – 2, change f(x) to y y = 8x – 2 Interchange x and y x = 8y – 2 Solve for y. Add 2 and divide by 8 ⅛x + ¼ = y ; so f -1 (x) = ⅛x + ¼

13. How would you check to make sure the functions are inverses? f(f -1 ) = 8(⅛x + ¼) – 2 = x + 2 – 2 = x f -1 (f) = ⅛(8x – 2) + ¼ = x - ¼ + ¼ = x

19. Would it change if y was y 2 ?

20. Homework Page 83 – 86 # 3, 9, 15, 23, 31, 43, 51, 63, 73, 91, 99

21. Homework Page 83 – 86 # 4, 11, 19, 29, 39, 47, 59, 69, 85, 95