1. Welcome to MM250 Unit 5 Seminar: Functions and Graphs To resize your pods: Place your mouse here. Left mouse click and hold. Drag to the right to enlarge the pod. To maximize chat, minimize roster by clicking here

2. Line y = 2x + 1 Function f(x) = 2x + 1

3. x -------------> ---------> f(x) f

4. Evaluating functions Ex: g(x) = 3x + 2 Then g(0) = g(1) = g(b) = g(x+4) =

5. Different types of Functions f(x) = x 2 + 1 f(x) = | x |

6. Different types of Functions Piecewise functions Ex: f(x) = x 2 when x <0 = 5x -1 when 0 ≤ x < 3 = 2x when x ≥ 3 What is f(-1), f(2), f(3)?

7. Relation a relation is a set of ordered pairs Ex: { (1,0), (3,2), (1,5), (6,8), (7,2) } Domain is the set of first elements {1, 3, 6, 7} Range is the set of second elements {0, 2, 5, 8}

8. Functions are special relations Ex: { (1,2), (1, 3), (3, 4), (7,8) } not a function Ex: { (1,3), (2,3), (5,7), (6,7) } function?

9. y = x 2 is a function every x goes to one y

10. x = y 2 does not define y as a function of x one x goes to two y's

11. Domain of a function Sometimes the domain of a function is stated. If not, we take the domain to be all real numbers that we can plug into the function that make sense. So, for the function f(x) = 1/x, what is the domain?

12. What is the domain of G(x) = √x

14. f(x) = x 4 - 5x 2 + 4

15. Odd and even functions A function is even is f(-x) = f(x) ex: f(x) = x 2 A function is odd if f(-x) = -f(x) ex: f(x) = x 3 A function is neither of none of the above apply g(x) = x 5 - x odd or even or neither?

16. f(x) = | x |

17. g(x) = f(x+2) = | x + 2 |

18. g(x) = f(x - 2) = | x - 2 |

19. g(x) = f(x) + 1 = | x | + 1

20. g(x) = f(x) - 1 = | x | - 1

21. g(x) = 3f(x) = 3| x |

22. Composition of functions (f o g)(x) = f( g(x) ) x ----> ----> g(x) -----> ---> f(g(x)) g f

23. Ex: let f(x) = 3x + 1 g(x) = x 2 Find (f o g)(x) Ex: let f(x) = 3 - x g(x) = x 3 Find (g o f)(2)

24. Inverse functions Sometimes you get x ----> ---------> ---> x f "undid" g g f

25. If both (f o g)(x) = x (g o f)(x) = x Then g is the inverse of f and we write g = f -1

26. Find the inverse of f(x) = 3x - 2