1. Functions…

2. Click on a Topic: • What is a Function? • Domains and Ranges • Meet the Parents • Transformations • Composition of Functions

3. The End (Now go practice what you’ve learned.

4. Function Compositions

5. So. . . how often do we reallyuse a composition of functions ?

6. Well, have you ever used the output from one operation on your calculator as the input of another?

7. I know I have. Me Too. Twice today! Then you have used a composition of functions.

8. Com-po-si-tion,n. Given two functions f and g, the composite function,

9. The Domain of the composition f(g(x)) is the set of all x values in the domain of g such that g(x) is in the domain of f.

10. Example: Find the domain of The domain of g(x): all reals of f(x):all reals except 0 of f(g(x)):all reals except -2

11. It is important to note that composition is not a commutative operation. Order does matter.

12. It is sometimes important to be able to “see” a function as a composition of two others. (There may be more than one possibility).

13. Find another f(x) and g(x) such that f(g(x)) = h(x)

14. Find another f(x) and g(x) such that f(g(x)) = h(x) Return to Topics

15. Transformations

16. Transformations allow us to take a “parent” function’s graph and move it up, down, left, or right stretch or shrink it vertically or horizontally, or flip it over an axis by means of algebraic changes to the function.

17. Translations of : • Slide UP K units • Slide DOWN K units Where k is a constant k > 0.

18. Parent:

19. To summarize, when a constant, k, is added or subtracted to the parent “after the fact” (outside of f(x) ) the result is a vertical translation up or down of y=f(x).

20. Translations of : • Slide LEFT K units • Slide RIGHT K units Where k is a constant k > 0.

21. Parent: Notice that subtracting moves it to the right and adding moves it to the left. This seems to be illogical. Explanation follows.

22. Remember: The notation f (x+k) or f (x-k) indicates that k is added to or subtracted from xbefore being evaluated in the function, f.

23. So, to get the same “output” value as y1 , the x substituted into y2 must be 3 less since we immediately add 3 to it. For y3 it must be 3 more because we immediately subtract 3 from it.

24. To summarize, when a constant, k, is added or subtracted to the x “before the fact” (inside of f(x) ) the result is a horizontal translation left or right of y=f(x) .

25. Reflections of : • Over the x axis • Over the y axis

26. Parent: Notice how the Range is affected in the graph on the left and how the Domain is affected in the graph on the right.

27. Reflections of : • Over the x axis • Over the y axis If a function is _______ then a reflection over the x axis is the same graph as a reflection over the y axis. odd

28. Reflections of : • Over the x axis • Over the y axis If a function is _______ then a reflection over the y axis leaves the graph unchanged. even

29. Reflections of : • Over the x axis • Over the y axis If a reflection over the x axis leaves the graph unchanged then____________________. the graph is not a function

30. Stretching/Shrinking: • Vertical Stretch • Vertical Shrink Where k is a constant k > 1.

31. Parent: Notice how the Range and amplitude (y data) are directly affected in each graph, however, Domain, and period (x data) are unaffected.

32. Stretching/Shrinking: • Horizontal Stretch • Horizontal Shrink Where k is a constant k > 1.

33. Parent: • Horizontal Stretch • Horizontal Shrink Notice how the Period is affected in each graph but Amplitude is not.

34. Notice: Dividing by k stretches the graph horizontally and multiplying by k shrinks the graph. Again, this may seem illogical, but. . .

35. Remember: The notation f (kx) or f (x/k) indicates that x is multiplied by k or divided by kbefore being evaluated in the function, f.

36. So, to get the same “output” value as y1 the x substituted into y2 must be 4times as big since we immediately divide it by 4 . For y3 it must be ¼ as big because we immediately multiply by 4. Return to Topics

39. What are “The Parents”? These are the 20 Basic functionsfrom which most of the functions and graphs that you will be working with this year will come. You need to know these thoroughly.

40. Line:

41. Absolute Value:

42. Quadratic:

43. Square Root

44. Cubic

45. Cube Root

46. Hyperbola:

47. Exponential:

48. Logarithmic:

49. Semicircle (Graph here “appears” to float above the axis, but it should not.)

50. Semi-hyperbola: (Graph here “appears” to float above the axis, but it should not.)