## anonymous one year ago Graph f(x)=x^2-3x-10/x-2. Find the domain, range, VA, HA/SA

1. misty1212

HI!!

2. anonymous

Hey:)

3. misty1212

is it $\frac{x^2-3x-10}{x-2}$

4. anonymous

Yes

5. misty1212

ok lets do them one at a time

6. misty1212

domain is easy enough, set the denominator equal to zero and solve then say "all real numbers except that one"

7. misty1212

what do you get for that? (you can do it in your head)

8. anonymous

2

9. misty1212

yeah so all numbers except 2

10. misty1212

now before we continue, lets make sure this is not a trick question can you factor the numerator and cancel?

11. anonymous

Let me see...

12. misty1212

hmmi think it does factor, but not cancel !

13. anonymous

(x-5)(x+2)/x-2

14. misty1212

yeah but nothing cancels, so lets continue but you should always check that first

15. misty1212

once we have the domain, we also have the vertical asymptote too

16. misty1212

clear or no?

17. anonymous

So it is also 2

18. anonymous

yes!

19. misty1212

careful here

20. misty1212

the vertical asymptote is a vertical line, not a number

21. misty1212

so you want to return and answer of $$x=2$$ the vertical line, not just the number 2

22. misty1212

clear?

23. anonymous

Oh! Yes! Clear

24. misty1212

now for the horizontal asympote the degree of the numerator is ?

25. anonymous

2

26. misty1212

and the denominator?

27. anonymous

there is no horizontal then since it's greater than the denom.

28. misty1212

bingo

29. misty1212

but the degree of the numerator is one more than the degree of the denominator that means there will be a slant asymptote do you know how to find it?

30. anonymous

I never learned about a slant asymptote so I dont know :(

31. misty1212

ok it is easy to say what to do , but not easy for me to write in one word "divide"

32. misty1212

when you divide a polynomial of degree 2 by a polynomial of degree 1 you get a polynomial of degree 1 ( a line) and a remainder ignore the remainder, the line is the slant asymptote

33. misty1212

you can use either long division or (much easier) synthetic division do you know how to do that?

34. anonymous

Yes :)

35. misty1212

whew like i said it is very hard for me to write division here what do you get?

36. misty1212

37. anonymous

I did it in a rush so I got x^2-1x-6.

38. misty1212

hmm no you should have a polynomial of degree 1, not a quadratic

39. misty1212

1 -3 -10 2 ______________ 1

40. misty1212

that is the iniital set up then 1 -3 -10 2 2 ______________ 1 -1

41. misty1212

the next line is unimportant since we only need the line, not the remainder the line has the coefficients from the final row so $$x-1$$ slant asymptote is $$y=x-1$$

42. anonymous

Oh! Now I know what I did wrong! When did the fak divison, i fogot to add the "0^2" Lol but yes, I know understand

43. misty1212

ok then we are done!

44. misty1212

oh except for the range

45. anonymous

I'm so bad at finding that.

46. misty1212

in this case it is real easy, since it has a vertical asympote and no horizontal asymptote that means it goes from $$-\infty$$ to $$\infty$$

47. misty1212

you can write your answer as an interval $$(-\infty, \infty)$$ or just say "all real numbers"

48. misty1212

it is harder for some other ones, but not for this one

49. anonymous

Yay! Thanks so much:)))

50. misty1212

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