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

- anonymous

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

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- misty1212

HI!!

- anonymous

Hey:)

- misty1212

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

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## More answers

- anonymous

Yes

- misty1212

ok lets do them one at a time

- misty1212

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

- misty1212

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

- anonymous

2

- misty1212

yeah so all numbers except 2

- misty1212

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

- anonymous

Let me see...

- misty1212

hmmi think it does factor, but not cancel !

- anonymous

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

- misty1212

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

- misty1212

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

- misty1212

clear or no?

- anonymous

So it is also 2

- anonymous

yes!

- misty1212

careful here

- misty1212

the vertical asymptote is a vertical line, not a number

- misty1212

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

- misty1212

clear?

- anonymous

Oh! Yes! Clear

- misty1212

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

- anonymous

2

- misty1212

and the denominator?

- anonymous

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

- misty1212

bingo

- 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?

- anonymous

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

- misty1212

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

- 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

- misty1212

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

- anonymous

Yes :)

- misty1212

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

- misty1212

take your time

- anonymous

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

- misty1212

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

- misty1212

1 -3 -10
2
______________
1

- misty1212

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

- 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 \(y=x-1\)

- 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

- misty1212

ok then we are done!

- misty1212

oh except for the range

- anonymous

I'm so bad at finding that.

- 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 \(\infty\)

- misty1212

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

- misty1212

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

- anonymous

Yay! Thanks so much:)))

- misty1212

\[\color\magenta\heartsuit\]

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