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Combined Functions

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- Two functions, f(x) and g(x), are combined in similar ways to addition, subtraction, multiplication, and division of real numbers. This allows the formation of new functions. - The document demonstrates examples of combining f(x)=x+1 and g(x)=2x using addition, subtraction, multiplication, and division. It shows the algebraic steps to derive each new function h(x) and evaluates h(x) for various x values. - Graphs of the functions h(x) are provided to illustrate the relationships between the combined functions.

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Combined Functions
•Two functions (f) and (g) are combined to form new functions in a similar manner that we add, subtract, multiply and divide real numbers
(f+g)(x) = (f)(g) (x) f(x) - g(x) f(x) ∙ g(x) f(x) g(x) f(x) + g(x) (f-g)(x) = (f ∙ g)(x) = f g (x) =
f(x) = x + 1 g(x) = 2x
f(x) = x + 1 g(x) = 2x f(x) + g(x) h(x) = f(x) + g(x) h(x) = (x + 1) + (2x) Substitute h(x) = (x + 1) + 1(2x) Imaginary 1 h(x) = x + 1 + 2x Add the same terms h(x) = 3x + 1
h(x) = 3x + 1h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 h(-2) = 3(-2) + 1 = -6+ 1 = -5 -5
h(x) = 3x + 1h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 h(-1) = 3(-1) + 1 = -3+ 1 = -2 -5 -2
h(x) = 3x + 1h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 h(0) = 3(0) + 1 = 0 + 1 = 1 -5 -2 1
h(x) = 3x + 1h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 h(1) = 3(1) + 1 = 3 + 1 = 4 -5 -2 1 4
h(x) = 3x + 1h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 h(2) = 3(2) + 1 = 6 + 1 = 7 -5 -2 1 4 7
GRAPH h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 -5 -2 1 4 7 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0
GRAPH h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 -5 -2 1 4 7 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0
GRAPH h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 -5 -2 1 4 7 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0
GRAPH h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 -5 -2 1 4 7 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0
GRAPH h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 -5 -2 1 4 7 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0
f(x) = x + 1 g(x) = 2x f(x) - g(x) h(x) = f(x) - g(x) h(x) = (x + 1) - (2x) Substitute h(x) = (x + 1) - 1(2x) Imaginary 1 h(x) = x + 1 -2x Subtract the same terms h(x) = -x + 1
h(x) = -x + 1h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 h(-2) = -(-2) + 1 = 2+ 1 = 3 3
h(x) = -x + 1h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 h(-1) = -(-1) + 1 = 1+ 1 = 2 3 2
h(x) = -x + 1h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 h(0) = -(0) + 1 = 0+ 1 = 1 3 2 1
h(x) = -x + 1h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 h(1) = -(1) + 1 = -1+ 1 = 0 3 2 1 0
h(x) = -x + 1h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 h(2) = -(2) + 1 = -2+ 1 = -1 3 2 1 0 -1
GRAPH h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 3 2 1 0 -1
GRAPH h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 3 2 1 0 -1
GRAPH h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 3 2 1 0 -1
GRAPH h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 3 2 1 0 -1
GRAPH h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 3 2 1 0 -1
f(x) = x + 1 g(x) = 2x f(x) ∙ g(x) h(x) = f(x) ∙ g(x) h(x) = (x + 1) ∙ (2x) Substitute h(x)= h(x) = [(2x)(x)] [(2x)(1)] Distribute 2x2 + 2x
h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 h(-2) = 2(-2)2 + 2(-2) = 4 4 h(x) = 2x2 + 2x = 2[(-2)(-2)] + 2(-2) = 2[4] - 4 = 8 - 4
h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 h(-1) = 2(-1)2 + 2(-1) = 0 4 h(x) = 2x2 + 2x = 2[(-1)(-1)] + 2(-1) = 2[1] - 2 = 2 - 2 0
h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 h(0) = 2(0)2 + 2(0) = 0 4 h(x) = 2x2 + 2x = 2[(0)(0)] + 2(0) = 2[0] - 0 = 0 - 0 0 0
h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 h(1) = 2(1)2 + 2(1) = 4 4 h(x) = 2x2 + 2x = 2[(1)(1)] + 2(1) = 2[1] + 2 = 2 + 2 0 0 4
h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 h(2) = 2(2)2 + 2(2) = 12 4 h(x) = 2x2 + 2x = 2[(2)(2)] + 2(2) = 2[4] + 4 = 8 + 4 0 0 4 12
GRAPH h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 4 0 0 4 12
GRAPH h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 4 0 0 4 12
GRAPH h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 4 0 0 4 12
GRAPH h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 4 0 0 4 12
GRAPH h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 4 0 0 4 12
f(x) = x + 1 g(x) = 2x h(x) = Substitute f(x) g(x) f(x) g(x) x + 1 2x h(x) =
x (f/g)(x) -2 -1 0 1 2 1/4 h(x) = X +1 2x x + 1 2x h(x) = -2 + 1 2(-2) h(-2) = - 1 - 4 = 1 4 =
x (f/g)(x) -2 -1 0 1 2 1/4 h(x) = X +1 2x x + 1 2x h(x) = -1 + 1 2(-1) h(-1) = 0 - 2 = = 0 0
x (f/g)(x) -2 -1 0 1 2 1/4 h(x) = X +1 2x x + 1 2x h(x) = 0 + 1 2(0) h(0) = 1 0 = = undefined 0 undefined
x (f/g)(x) -2 -1 0 1 2 1/4 h(x) = X +1 2x x + 1 2x h(x) = 1 + 1 2(1) h(1) = 2 2 = = 1 0 1 undefined
x (f/g)(x) -2 -1 0 1 2 1/4 h(x) = X +1 2x x + 1 2x h(x) = 2 + 1 2(2) h(2) = 3 4 = 0 1 3/4 undefined
GRAPH x (f/g)(x) -2 -1 0 1 2 -0.50-1 1 20 h(x) = X +1 2x 1/4 0 1 3/4 undefined -1.50-2 -0.50 -1.50 -1 -2 0.50 1 1.50 0.50 1.50 2
GRAPH x (f/g)(x) -2 -1 0 1 2 -0.50-1 1 20 h(x) = X +1 2x 1/4 0 1 3/4 undefined -2 -1.50 -0.50 -1.50 -1 -2 0.50 1 1.50 0.50 1.50 2
GRAPH x (f/g)(x) -2 -1 0 1 2 -0.50-1 1 20 h(x) = X +1 2x 1/4 0 1 3/4 undefined -2 -1.50 -1.50 -1 -2 0.50 1 1.50 0.50 1.50 2 -0.50
GRAPH x (f/g)(x) -2 -1 0 1 2 -0.50-1 1 20 h(x) = X +1 2x 1/4 0 1 3/4 undefined -2 -1.50 -0.50 -1.50 -1 -2 0.50 1 1.50 0.50 1.50 2
GRAPH x (f/g)(x) -2 -1 0 1 2 -0.50-1 1 20 h(x) = X +1 2x 1/4 0 1 3/4 undefined -2 -1.50 -0.50 -1.50 -1 -2 0.50 1 1.50 0.50 1.50 2
SOLVE FOR THE COMBINED FUNCTIONS, COMPLETE THEIR FUNCTION TABLES AND SKETCH THE GRAPH
GIVEN: f(x) = x - 3 g(x) = x2 SOLVE FOR: (f+g) (f-g) (f∙g)a. b. c. d. f g
Combined Functions

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Combined Functions

  • 2. •Two functions (f) and (g) are combined to form new functions in a similar manner that we add, subtract, multiply and divide real numbers
  • 3. (f+g)(x) = (f)(g) (x) f(x) - g(x) f(x) ∙ g(x) f(x) g(x) f(x) + g(x) (f-g)(x) = (f ∙ g)(x) = f g (x) =
  • 4. f(x) = x + 1 g(x) = 2x
  • 5. f(x) = x + 1 g(x) = 2x f(x) + g(x) h(x) = f(x) + g(x) h(x) = (x + 1) + (2x) Substitute h(x) = (x + 1) + 1(2x) Imaginary 1 h(x) = x + 1 + 2x Add the same terms h(x) = 3x + 1
  • 6. h(x) = 3x + 1h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 h(-2) = 3(-2) + 1 = -6+ 1 = -5 -5
  • 7. h(x) = 3x + 1h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 h(-1) = 3(-1) + 1 = -3+ 1 = -2 -5 -2
  • 8. h(x) = 3x + 1h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 h(0) = 3(0) + 1 = 0 + 1 = 1 -5 -2 1
  • 9. h(x) = 3x + 1h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 h(1) = 3(1) + 1 = 3 + 1 = 4 -5 -2 1 4
  • 10. h(x) = 3x + 1h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 h(2) = 3(2) + 1 = 6 + 1 = 7 -5 -2 1 4 7
  • 11. GRAPH h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 -5 -2 1 4 7 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0
  • 12. GRAPH h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 -5 -2 1 4 7 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0
  • 13. GRAPH h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 -5 -2 1 4 7 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0
  • 14. GRAPH h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 -5 -2 1 4 7 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0
  • 15. GRAPH h(x) = 3x + 1 x (f+g)(x) -2 -1 0 1 2 -5 -2 1 4 7 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0
  • 16. f(x) = x + 1 g(x) = 2x f(x) - g(x) h(x) = f(x) - g(x) h(x) = (x + 1) - (2x) Substitute h(x) = (x + 1) - 1(2x) Imaginary 1 h(x) = x + 1 -2x Subtract the same terms h(x) = -x + 1
  • 17. h(x) = -x + 1h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 h(-2) = -(-2) + 1 = 2+ 1 = 3 3
  • 18. h(x) = -x + 1h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 h(-1) = -(-1) + 1 = 1+ 1 = 2 3 2
  • 19. h(x) = -x + 1h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 h(0) = -(0) + 1 = 0+ 1 = 1 3 2 1
  • 20. h(x) = -x + 1h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 h(1) = -(1) + 1 = -1+ 1 = 0 3 2 1 0
  • 21. h(x) = -x + 1h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 h(2) = -(2) + 1 = -2+ 1 = -1 3 2 1 0 -1
  • 22. GRAPH h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 3 2 1 0 -1
  • 23. GRAPH h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 3 2 1 0 -1
  • 24. GRAPH h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 3 2 1 0 -1
  • 25. GRAPH h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 3 2 1 0 -1
  • 26. GRAPH h(x) = -x + 1 x (f-g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 3 2 1 0 -1
  • 27. f(x) = x + 1 g(x) = 2x f(x) ∙ g(x) h(x) = f(x) ∙ g(x) h(x) = (x + 1) ∙ (2x) Substitute h(x)= h(x) = [(2x)(x)] [(2x)(1)] Distribute 2x2 + 2x
  • 28. h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 h(-2) = 2(-2)2 + 2(-2) = 4 4 h(x) = 2x2 + 2x = 2[(-2)(-2)] + 2(-2) = 2[4] - 4 = 8 - 4
  • 29. h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 h(-1) = 2(-1)2 + 2(-1) = 0 4 h(x) = 2x2 + 2x = 2[(-1)(-1)] + 2(-1) = 2[1] - 2 = 2 - 2 0
  • 30. h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 h(0) = 2(0)2 + 2(0) = 0 4 h(x) = 2x2 + 2x = 2[(0)(0)] + 2(0) = 2[0] - 0 = 0 - 0 0 0
  • 31. h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 h(1) = 2(1)2 + 2(1) = 4 4 h(x) = 2x2 + 2x = 2[(1)(1)] + 2(1) = 2[1] + 2 = 2 + 2 0 0 4
  • 32. h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 h(2) = 2(2)2 + 2(2) = 12 4 h(x) = 2x2 + 2x = 2[(2)(2)] + 2(2) = 2[4] + 4 = 8 + 4 0 0 4 12
  • 33. GRAPH h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 4 0 0 4 12
  • 34. GRAPH h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 4 0 0 4 12
  • 35. GRAPH h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 4 0 0 4 12
  • 36. GRAPH h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 4 0 0 4 12
  • 37. GRAPH h(x) = 2x2 + 2x x (f∙g)(x) -2 -1 0 1 2 1 2 3 4 -1 -2 -3 -4 -5 -1-2-3-4 1 2 3 4 5 6 7 0 4 0 0 4 12
  • 38. f(x) = x + 1 g(x) = 2x h(x) = Substitute f(x) g(x) f(x) g(x) x + 1 2x h(x) =
  • 39. x (f/g)(x) -2 -1 0 1 2 1/4 h(x) = X +1 2x x + 1 2x h(x) = -2 + 1 2(-2) h(-2) = - 1 - 4 = 1 4 =
  • 40. x (f/g)(x) -2 -1 0 1 2 1/4 h(x) = X +1 2x x + 1 2x h(x) = -1 + 1 2(-1) h(-1) = 0 - 2 = = 0 0
  • 41. x (f/g)(x) -2 -1 0 1 2 1/4 h(x) = X +1 2x x + 1 2x h(x) = 0 + 1 2(0) h(0) = 1 0 = = undefined 0 undefined
  • 42. x (f/g)(x) -2 -1 0 1 2 1/4 h(x) = X +1 2x x + 1 2x h(x) = 1 + 1 2(1) h(1) = 2 2 = = 1 0 1 undefined
  • 43. x (f/g)(x) -2 -1 0 1 2 1/4 h(x) = X +1 2x x + 1 2x h(x) = 2 + 1 2(2) h(2) = 3 4 = 0 1 3/4 undefined
  • 44. GRAPH x (f/g)(x) -2 -1 0 1 2 -0.50-1 1 20 h(x) = X +1 2x 1/4 0 1 3/4 undefined -1.50-2 -0.50 -1.50 -1 -2 0.50 1 1.50 0.50 1.50 2
  • 45. GRAPH x (f/g)(x) -2 -1 0 1 2 -0.50-1 1 20 h(x) = X +1 2x 1/4 0 1 3/4 undefined -2 -1.50 -0.50 -1.50 -1 -2 0.50 1 1.50 0.50 1.50 2
  • 46. GRAPH x (f/g)(x) -2 -1 0 1 2 -0.50-1 1 20 h(x) = X +1 2x 1/4 0 1 3/4 undefined -2 -1.50 -1.50 -1 -2 0.50 1 1.50 0.50 1.50 2 -0.50
  • 47. GRAPH x (f/g)(x) -2 -1 0 1 2 -0.50-1 1 20 h(x) = X +1 2x 1/4 0 1 3/4 undefined -2 -1.50 -0.50 -1.50 -1 -2 0.50 1 1.50 0.50 1.50 2
  • 48. GRAPH x (f/g)(x) -2 -1 0 1 2 -0.50-1 1 20 h(x) = X +1 2x 1/4 0 1 3/4 undefined -2 -1.50 -0.50 -1.50 -1 -2 0.50 1 1.50 0.50 1.50 2
  • 49. SOLVE FOR THE COMBINED FUNCTIONS, COMPLETE THEIR FUNCTION TABLES AND SKETCH THE GRAPH
  • 50. GIVEN: f(x) = x - 3 g(x) = x2 SOLVE FOR: (f+g) (f-g) (f∙g)a. b. c. d. f g