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0210 ch 2 day 10

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The document discusses combining functions through addition, subtraction, multiplication, and division. Some key points: - Functions can be combined using the same rules as algebraic expressions, such as f(x) + g(x) = (f + g)(x). - An example demonstrates combining two functions f(x) and g(x) through addition, subtraction, multiplication, and division. - Composite functions are discussed where the output of one function acts as the input for another function, written as f(g(x)). - Several examples demonstrate evaluating composite functions by first evaluating the inner function and then the outer function. - It is shown that f(g(x)) is not

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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.”
We can add, subtract, multiply, divide functions
We can add, subtract, multiply, divide functions f (x) + g(x) = ( f + g)(x) D : D f I Dg
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
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
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
x−4 x−3 Example: f (x) = g(x) = x −1 x−2
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 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 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}
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} ⎝ ⎠
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
Composite Functions
Composite Functions The output of one function (the inner function) is used as input for another function (the outer function) notation : f ( g ( x ))
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
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
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
Composite Functions Example 2: y = sin ( 3θ )
Composite Functions Example 2: y = sin ( 3θ ) Inner 3θ g(θ ) = 3θ
Composite Functions Example 2: y = sin ( 3θ ) Inner 3θ g(θ ) = 3θ Outer sin ( x ) f (x) = sin ( x )
Composite Functions Example 2: y = sin ( 3θ ) Inner 3θ g(θ ) = 3θ Outer sin ( x ) f (x) = sin ( x ) f (g(θ )) = sin ( 3θ )
Composite Functions Example 3: y = sin 2 (θ )
Composite Functions Example 3: y = sin 2 (θ ) Inner sin (θ ) g(θ ) = sin (θ )
Composite Functions Example 3: y = sin 2 (θ ) Inner sin (θ ) g(θ ) = sin (θ ) Outer x 2 f (x) = x 2
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 )
Composite Functions Example 4: y = sin 2 ( 3θ )
Composite Functions Example 4: y = sin 2 ( 3θ ) Innermost 3θ h(θ ) = 3θ
Composite Functions Example 4: y = sin 2 ( 3θ ) Innermost 3θ h(θ ) = 3θ Next Inner sin (θ ) g(θ ) = sin (θ )
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
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 )
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
Is f (g(x)) = g( f (x)) ?
Is f (g(x)) = g( f (x)) ? Let f(x) = 2x+1 g(x) = x-3
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
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
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
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?
Could f (g(x)) = g( f (x)) ?
Could f (g(x)) = g( f (x)) ? x −1 Let f (x) = 2x + 1 and g(x) = 2
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
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
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.
HW #9 “There are precious few Einsteins among us. Most brilliance arises from ordinary people working together in extraordinary ways.” Roger Von Oech

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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
  • 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 )
  • 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

Editor's Notes