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1.7 Combination of Functions

1.7 Combination of Functions. Add, Subtract, Multiply and Divide functions Understanding combination functions. Sum and Difference. Let f(x) = 2x + 5 and g(x) = x 2 -3 Sum f(x) + g(x) = 2x + 5 + x 2 -3 thus (f + g)(x) = x 2 + 2x + 2

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1.7 Combination of Functions

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  1. 1.7 Combination of Functions Add, Subtract, Multiply and Divide functions Understanding combination functions

  2. Sum and Difference Let f(x) = 2x + 5 and g(x) = x2-3 Sum f(x) + g(x) = 2x + 5 + x2-3 thus (f + g)(x) = x2 + 2x + 2 Difference (f - g)(x) = 2x + 5 – (x2 – 3) = -x 2 + 2x + 8

  3. Product and Quotient Let f(x) = 2x + 5 and g(x) = x2-3 Product f(x)g(x) so (fg)(x) = (2x + 5)(x2 – 3) = 2x3 + 5x2 – 6x – 15 Quotient g(x)≠0

  4. So what is the domain of the Quotient function?

  5. So what is the domain of the Quotient function? All reals that does not make the denominator zero. What about a square root?

  6. Let and • The Domain of is [0, ∞) • The Domain of [ - 5, 5 ] Domain is [0, 5) ; Why?

  7. The zero comes from can not have negative numbers, • but must have numbers less then 5. Since it is in the denominator g(x) can not be zero. How would the Domain change if The Domain would be (0, 5] What cause the difference?

  8. Composition of Function Composition of function is where the range of one function become the domain of the other function. Let f(x) = x3 + 1 and g(x)= x + 4 Old way of written a composition was f(g(x)) New way (f∘g)(x) =(x+4)3 + 1 (g∘f)(x) = (x3+1)+4

  9. (f∘g)(x) =(x+4)3 + 1 (x + 4)3 = (x + 4)(x + 4)(x + 4) =(x2 + 8x + 16)(x + 4) =(x3 + 8x2 + 16x)+(4x2 + 32x + 64) = x3 + 12x2 + 48x + 64 (f∘g)(x) =x3 + 12x2 + 48x + 64 + 1 = x3 + 12x2 + 48x + 65

  10. Let f(x)= x2 + 8 and Domain of f(x) is ( - ∞,∞) Domain of g(x) is [- 4, 4] So the Domain would domain in which both equations work. [ -4, 4]

  11. Why use Composition functions? To break functions into smaller easier to handle parts. h(x)= (f∘g)(x) Into 2 equations f(x) = 1/x and g(x) =(x+ 3)2 or f(x) = (x + 3)-2 and g(x) = (x + 3)

  12. Homework • Page 74 – 76 # 1, 9, 19, 27, 33, 39, 43, 47, 51, 56, 60 ,67

  13. Homework • Page 74 – 76 # 5, 13, 22, 31, 35, 41, 45, 49, 53, 57,63

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