Finding the range of y = (x² - 4)/(x² - x - 12)

[GreenPrint]

Homework Statement

I'm stuck on this problem please help me

find the range of this equation

y = (x^2 - 4)/(x^2 - x - 12)

I know that it is [1/3,-infintiy]U[?,infinity]
yes I'm having trouble with that ?
At first I was like ok there should be a horizontal asymptote at y=1 there is in calculus deffinition the image as x approaches infinity from both sides right? or is just infinity... but you it appraoches one from both sides but it corsses one at x=-8 check the graph so how do I find how low it goes at this section also I can't use a calculator well I can but am suppose to show all work without using one i.e. have one to use not suppose to use it thanks

Homework Equations

The Attempt at a Solution

 

[QuarkCharmer]

Y = (x²-4)/(x²-x-12)

If you factor out the top and bottom of the equation you end up with

(x-2)(x+2) //Use the "difference of squares formula"
----------
(x+3)(x-4)

correct?

With no further simplifying to do, the only course of action you can really take is to figure out what Y cannot be right? What would the denominator have to equal, for the equation to have no solution?

 

[GreenPrint]

-3 and 4... but that wasn't my question I was asking for the range... please help me with that

 

[QuarkCharmer]

Ah, I apologise, I read that wrong.

Here is a good example of this problem.

Domain and Range
http://mathforum.org/library/drmath/view/53632.html

Domain, Range, and asymptote
http://mathforum.org/library/drmath/view/54477.html

 

[GreenPrint]

well I read them still can't solve the problem :(

 

[GreenPrint]

apparently I do this I don't understand what this guy is saying... or were he got .337 I got 1/3...

(-∞ , 0.337] U [ 0.969,∞)...look at the critical values and evaluate the function as well as doing the VA consideration ...and you certainly will need a numerical calculator to find the critical values and function values

 

[GreenPrint]

can someone please show me the calc to do this

 

[Dick]

QuarkCharmer gave you a hint. The numerator factors into (x+3)(x-4). So your function goes to infinity at x=(-3) and x=4. Use calculus to figure out what happens in the intervals (-infinity,-3), (-3,4) and (4,infinity) and put them together to get the total range.

 

[GreenPrint]

Can you please help me with this part... I have to use the chain rule right and the product rule to find the critical values correct then what do I do?

 

[Dick]

I think you just need the quotient rule to differentiate (x²-4)/(x²-x-12). Simplify it, set it equal to zero and solve for the critical points.

 

[GreenPrint]

what do I do after finding the critical points? Sorry i forgot how to do this stuff I should be able to find them eaisly but I don't remeber what to do after I find them...

 

[Dick]

It is kind of disappointing you don't know. The critical points are candidates for being maxes or mins on each interval. What you are basically doing is sketching a graph of y. Do that. Find the value at the critical points, find the limiting behavior at each side of x=-3 and x=4, and put that together with what you already know, that y=1 is a horizontal asymptote. Now draw a nice sketch of the function. Deduce the range from that.

 

[The Chaz]

Yeah, super disappointing.

After finding critical points, you could use a sign chart to determine which are maxima and which are minima

 

[GreenPrint]

ok well I posted this in the calc section sorry I posted it here didn't realize it would invovle calculus but you I found the cirtical values still couldn't solve the problem though even with a sign chart... please help me

 

[ehild]

If you know the critical points you can sketch the function. Find out how it behaves around -3 and 4. Where is it positive and where is it negative?

ehild

 

[GreenPrint]

I found them it didn't do me anything check other topic