How to determine domain and range

I seem to be confused on how to properly calculate domain and range.  Here are a few examples attached.  Can someone break this down for me in simple terms?  I have not done math for over 25 years and have forgotten most everything :(  Thanks
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wayy2beAsked:
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CompProbSolvCommented:
"How would you approach square roots in terms of domain and range?"

One method is what I suggested earlier.

f(X)=sqrt(X)

What values of X (in the real number system) don't give a valid answer for sqrt(X)?  The domain is everything in the real number system excluding those values.

Once you get that answer, look at what sqrt(X) will equal for that domain.  Try the lowest value of the domain, the largest value, and something inbetween and that will give you some clues.

"They are the numbers that I plugged in."
You plugged in 5/-20 for X/|X|.  That is using X=5 on top and X=-20 on the bottom.  X has to be the same in ALL places in the function.

That is:
f(5)=5/|5|
f(-20)=-20/|-20|

ozo made a VERY significant comment: "but you can't divide by 0".  That is one example of how you would exclude a number from the domain.  If a number would cause a function to have a divide by zero, that number is NOT in the domain!

That's not the only example, but a very significant one.
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ozoCommented:
Domain would be the possible values of x
Range would be the possible values of f(x)
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wayy2beAuthor Commented:
I need a much more detailed answer than that, working the problems.  I know the definitions but I need to understand how to apply them.
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ozoCommented:
For example, if we are dealing with real numbers, some functions make no sense for x=0, some functions make no sense for x < 0.
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d-glitchCommented:
Do you understand the rules about the sort of help you can get with homework and homework type question?

One of the best ways to understand these problems is to sketch the function.

You might note that the first one goes on forever, positive and negative, x and y.

The second one gets weird at zero.

The third has problems with negative numbers and zero.

And so on .....
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d-glitchCommented:
When you plot the 4th function, do you get a V or and upside down V?

Are there any problematic values for x?   Can you put in -100, 0, and +100?  How about -1000000  and  +1000000?

Are there any values of y that are not possible to generate?

When you answer those questions (for all four functions) you will know the domain and range.
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wayy2beAuthor Commented:
Thank you for your reply as well as your reminder concerning the rules.  This is not homework. This is a question related to domain and range of functions posted in Math & Science.
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CompProbSolvCommented:
My answer pretty much contains what has already been posted, but it might make more sense as it is just stated differently.

The domain is the range of values for X that give valid results for the function.  It should be obvious that it is easier to find what is NOT in the domain than what IS in the range.  So figure out all of the invalid (undefined) results, figure out what values of X gave you those results, and the range is everything except those values of X.

I'm assuming that you are dealing in the real number system, so "everything except" is the real number system less those values that you found that yield invalid results.

The range is the set of results that you will get for all of the values in the domain.  D-glitch makes a good suggestion to graph the function.  This should let you see what sorts of results you can get for each function.

An example of the domain and range for a different function may be helpful.  

Consider F(X)=X*X.  Is there any value for X (in the real number system) for which X*X gives an undefined value?  No.  Therefore the domain is -infinity<=X<=+infinity.

What will the graph of this function look like?  At -infinity the result is +infinity(squared).  At +infinity the result is also +infinity(squared).  At 0 the result is 0.  The function will look somewhat like the letter U with the base at (0,0).  The result never drops below 0, therefore the range is 0<=F(X)<=+infinity.  (I'm considering +infinity(squared) to be the same as +infinity here.)
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ozoCommented:
Our guidance might be better targeted if we can understand better what you find confusing about applying these concepts.
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wayy2beAuthor Commented:
Understood.  My issue is not understanding how to approach a problem.  Like this for example:

What is the range, R, of the function Image f(x) = x
                                                                                    ------
                                                                                    |x|

So this tells me that the range should be all positive numbers but what is the domain?
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ozoCommented:
Give that you cannot divide by 0, is there any x which cannot be used?

For the range, Is there any way to get f(x)=2?
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wayy2beAuthor Commented:
I don't believe so because 2 would always be divided by the absolute vale, yes?
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ozoCommented:
So if f(x) is never 2, then 2 is not in the range.

Can you think of any other numbers that are or are not in the range?
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wayy2beAuthor Commented:
1 and 0
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ozoCommented:
how do you get 0?
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d-glitchCommented:
f(x)  =  x / |x|  is a very interesting function.  

Try evaluating it at    x =  -2  -1  +1  +2  and  +100.
And think about evaluating it  x = 0
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wayy2beAuthor Commented:
Good question ozo!  That's where I get confused, what and where to plug in the values.... d-glitch maybe you can show me a worked out problem?
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ozoCommented:
for f(x)  =  x / |x|  you might start with noticing that |x| can be divided into distinct regions, within which it behaves more regularly, which can make the analysis simpler.
For positive x, |x| = x, so we have  f(x)  =  x / x for x > 0
For negative x, |x| = -x, so we have  f(x)  =  x / -x for x < 0
This leaves x=0, where |x| = 0, but you can't divide by 0, so 0 is not part of the domain of f(x)
So if you take all the possible values of x / x for x > 0, together with all the possible values of  x / -x for x < 0
you will have all the possible values of  f(x)
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wayy2beAuthor Commented:
Ok so 5/-20 = -15 so really any x/-x will be <0. Yes?
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ozoCommented:
5/-20 = -15
no, and and I'm not sure what 5/-20 has to do with x/|x|
x/-x will be <0
yes
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wayy2beAuthor Commented:
How would you approach square roots in terms of domain and range?
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wayy2beAuthor Commented:
no, and and I'm not sure what 5/-20 has to do with x/|x|

They are the numbers that I plugged in.
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ozoCommented:
When dealing with real numbers, the domain of sqrt(x) is x>=0, and the range is sqrt(x)>=0
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ozoCommented:
There is no x you can plug into x/|x| to get 5/-20, though it may be relevant to analyzing other functions, and 5/-20 = -0.25, not -15
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wayy2beAuthor Commented:
CompProbSolv: That post just made it extremely clear to me. I was plugging in all sorts of different numbers for top and bottom :S Thank you for setting me straight there.
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CompProbSolvCommented:
Glad to hear that it helped.

If different values for the top and bottom were allowed, they would have been different variables, such as:

f(X,Y)=X/|Y|

You would then have a domain for X, a domain for Y, and a two-dimensional range for X,Y.
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wayy2beAuthor Commented:
Thanks guys. I will I could give everyone points.
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