## moongazer Group Title how do you get the domain of this: one year ago one year ago

1. moongazer Group Title
2. moongazer Group Title

how do you find it? step-by-step please :)

3. ganeshie8 Group Title
4. moongazer Group Title

√(x^2 - 9x +20)/√(x^2 - 5x + 6) I already know the domain. But, how do you get it?

5. ganeshie8 Group Title

first workout contraints for squareroot : the thing inside squareroot cannot be negative

6. ganeshie8 Group Title

next work out constraints for denominator : denominator cannot be equal to 0

7. ganeshie8 Group Title

consider numerator first : $$\sqrt{x^2 - 9x +20}$$

8. ganeshie8 Group Title

can u work out domain for that ?

9. moongazer Group Title

{x|x>=5} is that right?

10. ganeshie8 Group Title

somewhat right :) lets see how to solve

11. ganeshie8 Group Title

x^2-9x+20 >= 0 x^2-4x-5x+20 >=0 x(x-4) -5(x-4) >= 0 (x-4)(x-5) >= 0 x <= 4 or x >= 5

12. ganeshie8 Group Title

thats one constraint for numerator, lets find out the constraints for denominator

13. moongazer Group Title

ohhhh

14. moongazer Group Title

I think I am starting to understand it :)

15. ganeshie8 Group Title

good :) since, denominator is also under radical, it must be >=0 : x^2 - 5x + 6 >= 0 x^2-3x-2x + 6 >= 0 x(x-3) -2(x-3) >= 0 (x-2)(x-3) >= 0 x <= 2 or x >= 3

16. ganeshie8 Group Title

with those two things, we finished with the first step of finding constraints for radicals

17. ganeshie8 Group Title

so far we have this : x <= 4 or x >= 5 x <= 2 or x >= 3

18. moongazer Group Title

yup, I understand it :)

19. ganeshie8 Group Title

next one is denominator can never equal to 0. so, x^2 - 5x + 6 $$\ne$$ 0 x^2-3x-2x + 6 $$\ne$$ 0 x(x-3) -2(x-3) $$\ne$$ 0 (x-2)(x-3) $$\ne$$ 0 x $$\ne$$ 2 or x $$\ne$$ 3

20. moongazer Group Title

w8 Isn't it that it should be: x >= 4 or x >= 5 x >= 2 or x >= 3 ???

21. ganeshie8 Group Title

lets combine all constraints and make a meaningful constraint

22. ganeshie8 Group Title

which one ?

23. moongazer Group Title

(x-4)(x-5) >= 0 (x-2)(x-3) >= 0

24. ganeshie8 Group Title

oh i got ur question, il give quick explanation

25. ganeshie8 Group Title

you comfortable with parabola graph ?

26. moongazer Group Title

I think so

27. ganeshie8 Group Title

il show u in graph why its x <=4 , x >=5

28. moongazer Group Title

I know few things about parabola

29. moongazer Group Title

ok

30. ganeshie8 Group Title

im sketching this parabola : (x-4)(x-5) >= 0

31. ganeshie8 Group Title

it intersect x axis at 4, 5 right ?

32. ganeshie8 Group Title

|dw:1348496600444:dw|

33. ganeshie8 Group Title

between 4 and 5, it is sinking down into x axis eh ?

34. ganeshie8 Group Title

its becoming NEGATIVE

35. ganeshie8 Group Title

|dw:1348496705657:dw|

36. moongazer Group Title

why did it intersect in x axis at 4, 5?

37. ganeshie8 Group Title

when u factored you got : (x-4)(x-5)

38. ganeshie8 Group Title

that means it intersects x-axis at 4 and 5

39. moongazer Group Title

ohh, ok what if it says (x-4)(x+5) does it intersect at 4 and -5 ?

40. ganeshie8 Group Title

thats right, when it says, (x-4)(x+5) = 0 then it intersects at 4 and -5

41. ganeshie8 Group Title

cuz for it to equal to 0, one of both of the factors must equal to 0

42. moongazer Group Title

|dw:1348498226449:dw|

43. moongazer Group Title

so the graph is like that?

44. ganeshie8 Group Title

perfect, thats (x-4)(x+5)

45. moongazer Group Title

and the domain is x>=4,x<=-5 is it correct?

46. ganeshie8 Group Title

thats right ! you became expert ;p

47. moongazer Group Title

for (x-4)(x+5)

48. moongazer Group Title

Thanks! I know understand it. :) let's now continue with our previous discussion :)

49. ganeshie8 Group Title

great ! so far what we have

50. moongazer Group Title

next one is denominator can never equal to 0. so, x^2 - 5x + 6 ≠ 0 x^2-3x-2x + 6 ≠ 0 x(x-3) -2(x-3) ≠ 0 (x-2)(x-3) ≠ 0 x ≠ 2 or x ≠ 3

51. ganeshie8 Group Title

yeah we almost done

52. ganeshie8 Group Title

lets put down all the 3 constraints we got in one place :

53. ganeshie8 Group Title

1) numerator radical : x <= 4 or x >= 5 2) denominator radical : x <= 2 or x >= 3 3) denominator cant be 0 : x ≠ 2 or x ≠ 3

54. moongazer Group Title

I understand it. What's next?

55. ganeshie8 Group Title

see 2) and 3)

56. ganeshie8 Group Title

can we combine them as one, like this : 2) 3) : x < 2 or x > 3

57. moongazer Group Title

I'll be back just continue.

58. moongazer Group Title

someone is calling me

59. ganeshie8 Group Title

ok after combining 2) and 3) we are left with : 1) numerator radical : x <= 4 or x >= 5 2) x < 2 or x > 3 this is the last step. see if u can figure out

60. ganeshie8 Group Title

out of x < 2 & x <= 4, x < 2 is more restrictive, so we have this in our domain : $$\color{green}{x < 2}$$

61. ganeshie8 Group Title

out of x > 3 & x <= 4 x > 3 , x <= 4 , which is same as : $$\color{green}{3<x\le 4}$$

62. ganeshie8 Group Title

and of course the last one, $$\color{green}{x >= 5}$$

63. moongazer Group Title

sorry for being away, I'm here again :)

64. ganeshie8 Group Title

np :) see if above stuff makes sense

65. moongazer Group Title

it makes sense but why in the final answer they used "or" not "and" "," "then" or anything else dom K: {x|x<2 or 3<x<=4 or x>=5}

66. ganeshie8 Group Title

yeah we always combine domain with "or" why ? cuz its meaningful. let me ask u a q

67. ganeshie8 Group Title

suppose we write : x < 2 and x >=5

68. moongazer Group Title

for the domain it is ALWAYS "or" ?

69. ganeshie8 Group Title

is that possible ? can x be BOTH less than 2 and greater than 5 at the SAME time ha ?

70. ganeshie8 Group Title

yes always "or"

71. moongazer Group Title

yes, it is not possible

72. ganeshie8 Group Title

is "1" less than 2 and greater than "5" ?

73. ganeshie8 Group Title

yeah, but if we put it like this : either "1" is less than 2 or greater than "5"

74. ganeshie8 Group Title

how meaningful it is, so while phrasing domain also, we use commonsense aswell :)

75. moongazer Group Title

yes, Thank you very much for helping me :)

76. ganeshie8 Group Title

np i learned as well. i forgot these long back in college... i enjoyed going thru these again.. thnks you too :)