Do you include the hole in the range of a rational function,?

[breed96]

So, for example, if I have:

x(x-2)^2
----------
x(x+2)

And I wanted to know where f(x) >0,


Would I say (-2,0)U(0,2)U(2,infinity),

or just

(-2,2)U(2,infinity)?



The only reason I am even asking this is because I don't know whether to factor out the hole (x=0) first, or leave it there.

 

[tkhunny]

In the Domain? Absolutely not.
In the Range? That depends on the would-be value being reproduced elsewhere.

Note: It's a line with two holes. What happened to \(\displaystyle -\infty\)?

 

[breed96]

In the Domain? Absolutely not.
In the Range? That depends on the would-be value being reproduced elsewhere.

The hole is defined as (0,2). Would that count as being reproduced elsewhere? How would you express the problem in my description?

 

[breed96]

In the Domain? Absolutely not.
In the Range? That depends on the would-be value being reproduced elsewhere.

Note: It's a line with two holes. What happened to \(\displaystyle -\infty\)?

none of the values below negative 2 produce anything > 0, and I'm not seeing any other factors shared on top and bottom

maybe I worded my question wrong. The description is a better description of my confusion lol

 

[breed96]

In the Domain? Absolutely not.
In the Range? That depends on the would-be value being reproduced elsewhere.

Note: It's a line with two holes. What happened to \(\displaystyle -\infty\)?

There aren't any other factors shared on top and bottom, and the problem asked for the values to be greater than 0

I think I worded the question wrong. My actual confusion is in the description

 

[breed96]

In the Domain? Absolutely not.
In the Range? That depends on the would-be value being reproduced elsewhere.

Note: It's a line with two holes. What happened to \(\displaystyle -\infty\)?

I think my question was a little misleading. The description is a more accurate representation of my confusion. Also, there was only one shared factor on top and bottom (hole)

 

[tkhunny]

It's a line. There is only one chance for each value. A hole will be missing from both Domain and Range.

 

[Dr.Peterson]

So, for example, if I have:

x(x-2)^2
----------
x(x+2)

And I wanted to know where f(x) >0,

Would I say (-2,0)U(0,2)U(2,infinity),

or just

(-2,2)U(2,infinity)?

The only reason I am even asking this is because I don't know whether to factor out the hole (x=0) first, or leave it there.

Let's start fresh. We need to be sure what the question really was.

Your title asks about the range; but the problem is not about a range. It is asking for the solution of
[x(x-2)^2]/[x(x+2)] > 0

Is that right?

It appears that you have simplified this by canceling the common factor, resulting in
(x-2)^2/(x+2)>0, x ≠ 0

Note that I included x ≠ 0 because canceling hides the fact that x can't be zero.

Then you solved this inequality to find that (ignoring the condition on x) the solution is
(-2,2) U (2,oo)

Combining everything, we end up with
(-2,0) U (0,2) U (2,oo)

Your answer is therefore correct. You can't include 0 because that makes the left side undefined.

Does this help? I think people misread a sign, so that they were making incorrect statements about the graph being a line, in addition to focusing on the question about range.