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This time

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  1. This time • (f+g)(x)=f(x)+g(x) • (f-g)(x)=f(x)-g(x) • (fg)(x)=f(x)*g(x) • (f/g)(x)=f(x)/g(x), g(x)≠0 • (f∘g)(x)=f(g(x))

  2. Things to remember • Function notation • ƒ(x)=2x-1 is a function definition • x is a number • ƒ(x) is a number • 2x-1 is a number • ƒ is the action taken to get from x to ƒ(x) • Multiply by 2 and add -1

  3. Things to remember • Function notation • ƒ(x)=2x-1 is a function definition • 3 is a number • ƒ(3) is a number • 2*3-1 is a number (it’s 5) • ƒ is the action taken to get from 3 to ƒ(3) • Multiply by 2 and add -1

  4. Practice using notation

  5. I can do a function to a number

  6. What’s going on here?

  7. Composing functions algebraically

  8. In diagram form

  9. WARNING • Parentheses are ambiguous • When you have twoNUMBERS, a(b) means “multiply a and b” • When you have aFUNCTION, a(b) means “do the action called a to the number b.” • Always keep track of what’s a function and what’s a number.

  10. The most common confusion of all time • (f+g)(x)=f(x)+g(x) • (f-g)(x)=f(x)-g(x) • (fg)(x)=f(x)*g(x) • (f/g)(x)=f(x)/g(x), g(x)≠0 • (f∘g)(x)=f(g(x))

  11. COMPARISON (f∘g)(3)=f(g(3)) (fg)(3)=f(3)g(3) -1≠0

  12. WARNING (fg)(x) and f(g(x)) are not the same thing • (fg)(x) means “do f to x, then do g to x, then multiply the numbers f(x) and g(x).” • f(g(x)) means “do g to x, get the number g(x), then do f to the number g(x)” • No multiplying.

  13. In picture form f(x) f f(x)g(x) x * g g(x) Is not the same as g f x g(x) f(g(x))

  14. Interpretive Dance All about multiplication

  15. Time to dance!

  16. Add 1 to each number

  17. Add -1 to each number

  18. Multiply each number by 0.5

  19. Multiply each number by 2

  20. Multiplication is not repeated addition • Addition is shifting • Multiplication is stretching • And shrinking • No amount of repeated shifting will give you a stretch

  21. Transformations of functions And their graphs

  22. This is a graph of a function called ƒ

  23. Let g(x)=ƒ(x)+1.5

  24. What does a graph of g look like?

  25. g(3)=ƒ(3)+1.5 (3,f(3)+1.5) (3,f(3))

  26. g(3)=ƒ(3)+1.5 (3,f(3)+1.5) The x is still the same, But the y is 1.5 higher (3,f(3))

  27. Draw a graph where all the x’s are the same and all the y’s are 1.5 higher. (3,f(3)+1.5) The x is still the same, But the y is 1.5 higher (3,f(3))

  28. The graph of f(x)+1.5 is the graph of f(x) shifted up by 1.5 (3,f(3)+1.5) The x is still the same, But the y is 1.5 higher (3,f(3))

  29. Draw a graph where all the x’s are the same and all the y’s are 1.5 higher. (3,f(3)+1.5) The x is still the same, But the y is 1.5 higher (3,f(3))

  30. Graphing Transformations • The graph for ƒ(x)+c is the graph of ƒ(x) shifted up by c.

  31. This is a graph of a function called ƒ

  32. Let g(x)=ƒ(x-1)

  33. What does a graph of g look like?

  34. g(3)=ƒ(3-1)=ƒ(2) (2,f(2)) (3,f(2))

  35. g(3)=ƒ(3-1)=ƒ(2) The y of g(3) is the same as the y of f(2) (2,f(2)) (3,f(2))

  36. g(3)=ƒ(3-1)=ƒ(2) The y of g(3) is the same as the y of f(2) Thinking from ƒg, The y is the same, but the x needed to make that y is 1 bigger (2,f(2)) (3,f(2))

  37. Draw a graph where the y’s are the same, but you need a 1 bigger x to make each one. The y of g(3) is the same as the y of f(2) Thinking from ƒg, The y is the same, but the x needed to make that y is 1 bigger (2,f(2)) (3,f(2))

  38. Graphing Transformations • The graph for ƒ(x)+c is the graph of ƒ(x) shifted up by c. • The graph for ƒ(x-a) is the graph of ƒ(x) shifted right by a. • NOTE THE MINUS SIGN

  39. This is a graph of a function called ƒ

  40. Let g(x)=2ƒ(x)

  41. g(-1.5)=2ƒ(-1.5) (-1.5,f(-1.5)) (-1.5,2f(-1.5))

  42. g(-1.5)=2ƒ(-1.5) The x is still the same, But the y is twice as far away from zero (-1.5,f(-1.5)) (-1.5,2f(-1.5))

  43. Draw a graph where the x’s stay the same, but the y’s are twice as far away from zero. The x is still the same, But the y is twice as far away from zero (-1.5,f(-1.5)) (-1.5,2f(-1.5))

  44. Graphing Transformations • The graph for ƒ(x)+c is the graph of ƒ(x) shifted up by c. • The graph for ƒ(x-a) is the graph of ƒ(x) shifted right by a. • NOTE THE MINUS SIGN • The graph for rƒ(x) is the graph of ƒ(x) stretched vertically by r.

  45. This is a graph of a function called ƒ

  46. Let g(x)=ƒ(2x)

  47. g(1.5)=ƒ(2*1.5)=ƒ(3) The y is still the same, but the x needed to make that y is half as big (1.5,f(3)) (3,f(3))

  48. Draw a graph where the y’s stay the same, but the x’s needed to make those graphs are half as big. The y is still the same, but the x needed to make that y is half as big (1.5,f(3)) (3,f(3))

  49. Graphing Transformations • The graph for ƒ(x)+c is the graph of ƒ(x) shifted up by c. • The graph for ƒ(x-a) is the graph of ƒ(x) shifted right by a. • NOTE THE MINUS SIGN • The graph for rƒ(x) is the graph of ƒ(x) stretched vertically by r. • The graph for ƒ(sx) is the graph of ƒ(x) squished horizontally by s.

  50. Graphing Transformations • The graph for ƒ(x)+c is the graph of ƒ(x) shifted up by c. • The graph for ƒ(x-a) is the graph of ƒ(x) shifted right by a. • NOTE THE MINUS SIGN • The graph for rƒ(x) is the graph of ƒ(x) stretched vertically by r. • The graph for ƒ(sx) is the graph of ƒ(x) squished horizontally by s. • Note the difference!