# Homework Help: Computing the range for a rational function involving absolute value

1. Nov 3, 2006

### Checkfate

Hi. I need help computing a range.

The question is : Find the domain and range of $$y=\frac{|x+2|}{x}$$.

The domain is obvious, x can't be 0, (-inf,0,) U (0,inf). But how do I find the range?? Can someone help me out? I have tried messing around with the definition of absolute value... if x>0 then |x| = x and if x<0 then |x| = -(x) but it just adds to the confusion, it probably doesn't help that I am running on like 4 cups of coffee. Anyways, thanks for the help

2. Nov 3, 2006

### nazzard

Hello Checkfate,

you can continue to use the definition of absolute value and rewrite the function for both cases. For $x\geq-2$ you would get
$$y=\frac{|x+2|}{x}=\frac{x+2}{x}=1+\frac{2}{x}$$
and you can discuss the range of this function for $x\in[-2,0)$ and $x\in(0,\infty)$.

(similar for the case of $x<-2$)

Regards,

nazzard

Last edited: Nov 3, 2006
3. Nov 3, 2006

### Checkfate

Okay, I came to the right answer.... this is how I did it

I started as you suggested, by first working with the function as defined when $x\geq-2$ and got the equation $$y=1+\frac{2}{x}$$. I then had to do some thinking... in the interval [-2,0) y starts out at a maximum of 0 and then quickly declines from that point on as $$\frac{2}{x}$$ gets larger and larger, in a negative fashion. Then after 0, $$\frac{2}{x}$$ starts out infinitely large and comes down and tends to a minimum of 1 as x gets larger. So the range for that interval is $$(-\inf,0] U (1,\inf)$$... Then I look at when $$x<-2$$ and note that as x decreases, $$\frac{-2}{x}$$ gets closer to zero and this the function tends towards -1. So the range in this interval is $$(-1,0}$$... This gives a combined range of $$(-\inf,0] U (1,\inf)$$.

Thanks alot :)

Last edited: Nov 3, 2006
4. Nov 3, 2006

### arildno

Why is the domain obvious?
I don't get that at all.
Please tell me.

5. Nov 4, 2006

### Checkfate

Because with |x+2|/x, the only value of x that will mess it up is 0... Division by zero is undefined :P Everything else is fair game.

6. Nov 4, 2006

### arildno

Well, why can't its domain be [2,4], then?
Or the set of points $\{-1,3.14,57\}$?

It is meaningless to supply a function without specifying its domain.
What you have specified, is known as the MAXIMAL domain of the function within the real number set.

Last edited: Nov 4, 2006
7. Nov 4, 2006

### Checkfate

But isn't agreeable that when you are asked to specify the domain of a function, what is being asked for is all the possible x's that can be plugged into the function? Unless you are supplied with a graph of the function, then you could only go on what you see.

I do see your point though :)

8. Nov 5, 2006

### arildno

It is obvious that that is what the dumb exercise maker MEANT.
However, that does not excuse the exercise maker for making an improper question!

He should have asked something like:
"Determine the greatest set of real numbers that can serve as the domain of the function"

9. Mar 13, 2010

### FlO-rida

i have to do a portfolio on rational functions, so i am supposed to figure out everuthing about asymptotes, range, domain, etc. all by my self. so far i have been succesful in figuring out the domain but still dont know how to find the range of a rational function. for ex. 1/x+3

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