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0210 ch 2 day 10
- 1. 2.7 Combining Functions John 16:33 I have said these things to you, that in me you may have peace. In the world you will have tribulation. But take heart; I have overcome the world.”
- 2. We can add, subtract, multiply, divide functions
- 3. We can add, subtract, multiply, divide functions f (x) + g(x) = ( f + g)(x) D : D f I Dg
- 4. We can add, subtract, multiply, divide functions f (x) + g(x) = ( f + g)(x) D : D f I Dg f (x) − g(x) = ( f − g)(x) D : D f I Dg
- 5. We can add, subtract, multiply, divide functions f (x) + g(x) = ( f + g)(x) D : D f I Dg f (x) − g(x) = ( f − g)(x) D : D f I Dg f (x)⋅ g(x) = ( f ⋅ g)(x) D : D f I Dg
- 6. We can add, subtract, multiply, divide functions f (x) + g(x) = ( f + g)(x) D : D f I Dg f (x) − g(x) = ( f − g)(x) D : D f I Dg f (x)⋅ g(x) = ( f ⋅ g)(x) D : D f I Dg f (x) ⎛ f ⎞ D : D f I Dg = ⎜ ⎟ ( x ) g(x) ⎝ g ⎠ and g(x) ≠ 0
- 7. x−4 x−3 Example: f (x) = g(x) = x −1 x−2
- 8. x−4 x−3 Example: f (x) = g(x) = x −1 x−2 x−4 x−3 ( f + g)(x) = + D: {x : x ≠ 1,2} x −1 x − 2
- 9. x−4 x−3 Example: f (x) = g(x) = x −1 x−2 x−4 x−3 ( f + g)(x) = + D: {x : x ≠ 1,2} x −1 x − 2 x−4 x−3 ( f − g)(x) = − D: {x : x ≠ 1,2} x −1 x − 2
- 10. x−4 x−3 Example: f (x) = g(x) = x −1 x−2 x−4 x−3 ( f + g)(x) = + D: {x : x ≠ 1,2} x −1 x − 2 x−4 x−3 ( f − g)(x) = − D: {x : x ≠ 1,2} x −1 x − 2 ⎛ x − 4 ⎞ ⎛ x − 3 ⎞ ( f ⋅ g)(x) = ⎜ ⎝ x − 1 ⎟ ⎜ x − 2 ⎟ ⎠ ⎝ ⎠ D: {x : x ≠ 1,2}
- 11. x−4 x−3 Example: f (x) = g(x) = x −1 x−2 x−4 x−3 ( f + g)(x) = + D: {x : x ≠ 1,2} x −1 x − 2 x−4 x−3 ( f − g)(x) = − D: {x : x ≠ 1,2} x −1 x − 2 ⎛ x − 4 ⎞ ⎛ x − 3 ⎞ ( f ⋅ g)(x) = ⎜ ⎝ x − 1 ⎟ ⎜ x − 2 ⎟ ⎠ ⎝ ⎠ D: {x : x ≠ 1,2} ⎛ f ⎞ ⎛ x − 4 ⎞ ⎛ x − 2 ⎞ ⎜ g ⎟ (x) = ⎜ x − 1 ⎟ ⎜ x − 3 ⎟ ⎝ ⎠ ⎝ ⎠ D: {x : x ≠ 1,2, 3} ⎝ ⎠
- 12. x−4 x−3 Example: f (x) = g(x) = x −1 x−2 x−4 x−3 ( f + g)(x) = + D: {x : x ≠ 1,2} x −1 x − 2 x−4 x−3 ( f − g)(x) = − D: {x : x ≠ 1,2} x −1 x − 2 ⎛ x − 4 ⎞ ⎛ x − 3 ⎞ ( f ⋅ g)(x) = ⎜ ⎝ x − 1 ⎟ ⎜ x − 2 ⎟ ⎠ ⎝ ⎠ D: {x : x ≠ 1,2} ⎛ f ⎞ ⎛ x − 4 ⎞ ⎛ x − 2 ⎞ ⎜ g ⎟ (x) = ⎜ x − 1 ⎟ ⎜ x − 3 ⎟ ⎝ ⎠ ⎝ ⎠ D: {x : x ≠ 1,2, 3} ⎝ ⎠ Be sure to read Example 2 in your textbook
- 14. Composite Functions The output of one function (the inner function) is used as input for another function (the outer function) notation : f ( g ( x ))
- 15. Composite Functions The output of one function (the inner function) is used as input for another function (the outer function) notation : f ( g ( x )) Example 1: y = x +1
- 16. Composite Functions The output of one function (the inner function) is used as input for another function (the outer function) notation : f ( g ( x )) Example 1: y = x +1 Inner function is done first x +1 Outer function is done second x
- 17. Composite Functions The output of one function (the inner function) is used as input for another function (the outer function) notation : f ( g ( x )) Example 1: y = x +1 Inner function is done first x +1 Outer function is done second x g(x) = x + 1 f (x) = x f (g(x)) = x + 1
- 18. Composite Functions Example 2: y = sin ( 3θ )
- 19. Composite Functions Example 2: y = sin ( 3θ ) Inner 3θ g(θ ) = 3θ
- 20. Composite Functions Example 2: y = sin ( 3θ ) Inner 3θ g(θ ) = 3θ Outer sin ( x ) f (x) = sin ( x )
- 21. Composite Functions Example 2: y = sin ( 3θ ) Inner 3θ g(θ ) = 3θ Outer sin ( x ) f (x) = sin ( x ) f (g(θ )) = sin ( 3θ )
- 22. Composite Functions Example 3: y = sin 2 (θ )
- 23. Composite Functions Example 3: y = sin 2 (θ ) Inner sin (θ ) g(θ ) = sin (θ )
- 24. Composite Functions Example 3: y = sin 2 (θ ) Inner sin (θ ) g(θ ) = sin (θ ) Outer x 2 f (x) = x 2
- 25. Composite Functions Example 3: y = sin 2 (θ ) Inner sin (θ ) g(θ ) = sin (θ ) Outer x 2 f (x) = x 2 f (g(θ )) = sin 2 (θ )
- 26. Composite Functions Example 4: y = sin 2 ( 3θ )
- 27. Composite Functions Example 4: y = sin 2 ( 3θ ) Innermost 3θ h(θ ) = 3θ
- 28. Composite Functions Example 4: y = sin 2 ( 3θ ) Innermost 3θ h(θ ) = 3θ Next Inner sin (θ ) g(θ ) = sin (θ )
- 29. Composite Functions Example 4: y = sin 2 ( 3θ ) Innermost 3θ h(θ ) = 3θ Next Inner sin (θ ) g(θ ) = sin (θ ) Outer x 2 f (x) = x 2
- 30. Composite Functions Example 4: y = sin 2 ( 3θ ) Innermost 3θ h(θ ) = 3θ Next Inner sin (θ ) g(θ ) = sin (θ ) Outer x 2 f (x) = x 2 f (g(h(θ ))) = sin 2 ( 3θ )
- 31. Composite Functions Example 4: y = sin 2 ( 3θ ) Innermost 3θ h(θ ) = 3θ Next Inner sin (θ ) g(θ ) = sin (θ ) Outer x 2 f (x) = x 2 f (g(h(θ ))) = sin 2 ( 3θ ) Be sure to read Example 7 in your textbook
- 32. Is f (g(x)) = g( f (x)) ?
- 33. Is f (g(x)) = g( f (x)) ? Let f(x) = 2x+1 g(x) = x-3
- 34. Is f (g(x)) = g( f (x)) ? Let f(x) = 2x+1 g(x) = x-3 f (g(x)) = 2(x − 3) + 1 = 2x − 6 + 1 = 2x − 5
- 35. Is f (g(x)) = g( f (x)) ? Let f(x) = 2x+1 g(x) = x-3 f (g(x)) = 2(x − 3) + 1 g( f (x)) = (2x + 1) − 3 = 2x − 6 + 1 = 2x − 2 = 2x − 5
- 36. Is f (g(x)) = g( f (x)) ? Let f(x) = 2x+1 g(x) = x-3 f (g(x)) = 2(x − 3) + 1 g( f (x)) = (2x + 1) − 3 = 2x − 6 + 1 = 2x − 2 = 2x − 5 Not the same
- 37. Is f (g(x)) = g( f (x)) ? Let f(x) = 2x+1 g(x) = x-3 f (g(x)) = 2(x − 3) + 1 g( f (x)) = (2x + 1) − 3 = 2x − 6 + 1 = 2x − 2 = 2x − 5 Not the same Could they be the same?
- 38. Could f (g(x)) = g( f (x)) ?
- 39. Could f (g(x)) = g( f (x)) ? x −1 Let f (x) = 2x + 1 and g(x) = 2
- 40. Could f (g(x)) = g( f (x)) ? x −1 Let f (x) = 2x + 1 and g(x) = 2 ⎛ x − 1 ⎞ f (g(x)) = 2 ⎜ ⎟ + 1 ⎝ 2 ⎠ = x − 1+ 1 =x
- 41. Could f (g(x)) = g( f (x)) ? x −1 Let f (x) = 2x + 1 and g(x) = 2 ⎛ x − 1 ⎞ (2x + 1) − 1 f (g(x)) = 2 ⎜ ⎟ + 1 g( f (x)) = ⎝ 2 ⎠ 2 2x = x − 1+ 1 = 2 =x =x
- 42. Could f (g(x)) = g( f (x)) ? x −1 Let f (x) = 2x + 1 and g(x) = 2 ⎛ x − 1 ⎞ (2x + 1) − 1 f (g(x)) = 2 ⎜ ⎟ + 1 g( f (x)) = ⎝ 2 ⎠ 2 2x = x − 1+ 1 = 2 =x =x These are the same. It happens when f(x) and g(x) are inverses of each other.
- 43. HW #9 “There are precious few Einsteins among us. Most brilliance arises from ordinary people working together in extraordinary ways.” Roger Von Oech