find the domain of the rational function.

Question

anonymous · Answer

$$f(y)=\frac{ y+8 }{ y ^{2} -8y}$$

e.mccormick · Answer

Do you know what the concept of domain restrictions is?

anonymous · Answer

i completely forgot it.

e.mccormick · Answer

Whenever the equation would become invalid, or not exist, those values can not be part of the domain.  In a rational equation, that is when the bottom of the faction would be 0.

anonymous · Answer

$$\frac{ y+8 }{ 0}$$

e.mccormick · Answer

Yes, any place that would happen would make it invalid,.  So when is the bottom zero?
$y^2-8y=0$

anonymous · Answer

okay, but how do I solve this.... I'm so confused

e.mccormick · Answer

Factor it.  Set each factor to 0.  Solve for y on each factor.

anonymous · Answer

can you show me an example?

e.mccormick · Answer

$x^2+2x-3=(x+3)(x-1)$ That is the factoring part.  Getting each piece to a multiplied little bit with a single variable.  then set each factor to 0 and solve.
$x+3=0\implies x=-3$
$x-1=0\implies x=1$

anonymous · Answer

(y-+8)(y-1)?

e.mccormick · Answer

What if that was the bottom oart of a fraction? To show you why it would be invalid, look at this graph: https://www.desmos.com/calculator/pl8orqyjka See where $x=-3$ and $x-1$ are?

e.mccormick · Answer

Not quite.  That would not multiply back to yours.  When there are just two parts, and they both have a value, it is best to factor out that value.  Then you can see if there is more that could be done.  In $y^2-8y$, do you see a common value to both of those?

anonymous · Answer

would it be Y?

anonymous · Answer

sigh! AH! i'm so lost right now.

e.mccormick · Answer

Yes.  So if you factor out a y, what would you get?  For example: $10+2=2(5+1)$

anonymous · Answer

y(y-8)?

e.mccormick · Answer

Exactly!  So, those are the factors in this case, y and y-8.  What do you get if you set each of those to 0?

anonymous · Answer

-8 and 8?

anonymous · Answer

or 8y?

e.mccormick · Answer

If you set them to 0, like I did, you get:
$y=0$
$y-8=0$

Now, what do you notice about the first one.  Ignore the second for a moment.

e.mccormick · Answer

Remember, the goal is to solve for possible values of y, so we want y=something.

anonymous · Answer

its 0 and you cant divide with anything?

e.mccormick · Answer

Exactly.  0 is one of the points that is a problem.  So that is ONE of the answers.  Write it down.  Now we move on to the second one:

$y-8=0$

What do we need to do to make that into y=something?

anonymous · Answer

add 8 to both sides?

e.mccormick · Answer

EXACTLY!
$y=8$ so... guess what that means.

anonymous · Answer

y=8

anonymous · Answer

my answers are 8 and 0

e.mccormick · Answer

is a?

e.mccormick · Answer

Yes!

anonymous · Answer

so how do i write that in interval notation?

e.mccormick · Answer

Now we have two factors and two answers, so there is no more to be done math wise.
The equation would be invalid any place the bottom could be 0.  The bottom can be zero at $y=\{0,8\}$

Now, it just needs to be written properly.

e.mccormick · Answer

Interval notation:
( and ) mean start/end here but do not include this value.
[ and ] mean start/end here but do include this value.

Because these are values that are not included, we will use ( ).

e.mccormick · Answer

The minimum value is $-\infty$ and the maximum is $\infty$.  They can never be included.

e.mccormick · Answer

$\cup$ means union, so in this or in that.

anonymous · Answer

okay.

e.mccormick · Answer

So, now that you have the basics... let me show you what I mean.

$(-\infty,Something)\cup(next,\infty)$ is the very basic form.  In your case, it will be a tad diffeerent because you have two invalid numbers, and stuff between them is valid.

e.mccormick · Answer

Here, this might be clearer.  If I said, "Everything except 1" it would be this in interval notation:
$(-\infty,1)\cup(1,\infty)$

e.mccormick · Answer

If I said everything except 1 and 5, it becomes this:
$(-\infty,1)\cup(1,5)\cup(5,\infty)$

anonymous · Answer

so my answer would look like (−∞,0)∪(0,8)∪(8,∞)

e.mccormick · Answer

Yes!

e.mccormick · Answer

Just for reference on the [ ] part, if I said everything from 5 and above, including 5, it would be this:
$[5,\infty)$

anonymous · Answer

so i need to include the bracket in front of my 8?

e.mccormick · Answer

No, 8 is NOT included on yours.  I was just showing you that for reference.

anonymous · Answer

ohhh ok! LOL!

e.mccormick · Answer

Now, you know how on a graph the x is right to left and the y is up and down? Well, here is the graph of your question: https://www.desmos.com/calculator/ingjpjapa9 See where y=0 and and y=8 are at? Notice how it just curves away from there and never makes it?

anonymous · Answer

OMG!!! I cannot thank you enough for your time and patience with me.

e.mccormick · Answer

If this equation was some path you were trying to follow, those points, 0 and 8, would be the ultimate road blocks.  They go on forever and you can never touch them.

anonymous · Answer

I already bookmarked that bad boy

anonymous · Answer

yes, when I entered it I noticed that.

e.mccormick · Answer

Yah, a good graph is a wonderful tool.  Try and solve things, and get the math down, but the graph makes a great check to make sure you got it right.

e.mccormick · Answer

Here is another good resource for you. http://www.purplemath.com/modules/index.htm they take each topic by name. Find things you have trouble with, like the factoring we did, and review them on there. If you deal with just one concept at a time until you remember the basics, then it is not so hard.

anonymous · Answer

awesome!! bookmarked as well.

e.mccormick · Answer

There are a few very good math sites out there. I just like how they break it down on that one.  Makes it easier and their topics list is not hard to search.  Just use the find thing in your browser, like ctrl-f key combination, then type in the name of a topic in your book, and there it is!

anonymous · Answer

me too! i was browsing through it now. great info!

e.mccormick · Answer

The explanations also tend to be better than most books.  They take the time on complex topics to do 4 or 5 pages, where I have seen the same topic in a math book in 2 pages.

e.mccormick · Answer

For example, look at Composition: http://www.purplemath.com/modules/fcncomp.htm 6 pages there. Probably twice the information as in a regular math book. Lots of fuly worked examples.

anonymous · Answer

yes!!! I saw that... thanks so much for your help and time!! it is much appreciated.

anonymous · Answer

i am going to close this question, cause I have another equation I am stuck on. :(

e.mccormick · Answer

np.  Don't have too much fun!

Oh, and one other tip.  When I go into a class, I find college produced videos on Youtube covering the topic, and preview it while I am on break.  That way I have an idea about things before class starts and I am less lost with terms and new things.

anonymous · Answer

math and i arent the bestest friends. Haaa!! thank you sir. xoxo