Computing the range for a rational function involving absolute value

[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

 

[nazzard]

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??

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

 

[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 a lot :)

 

[arildno]

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

 

[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.

 

[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.

 

[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 :)

 

[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"

 

[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 successful in figuring out the domain but still don't know how to find the range of a rational function. for ex. 1/x+3