Computing the range for a rational function involving absolute value

In summary: 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,you can use the definition of absolute value to find the range for x\in(-2,0) and x\in(0,\infty).
  • #1
Checkfate
149
0
Hi. I need help computing a range.

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

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 :approve:
 
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  • #2
Checkfate said:
Hi. I need help computing a range.

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

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 [itex]x\geq-2[/itex] you would get
[tex]y=\frac{|x+2|}{x}=\frac{x+2}{x}=1+\frac{2}{x}[/tex]
and you can discuss the range of this function for [itex]x\in[-2,0)[/itex] and [itex]x\in(0,\infty)[/itex].

(similar for the case of [itex]x<-2[/itex])

Regards,

nazzard
 
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  • #3
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 [itex]x\geq-2[/itex] and got the equation [tex]y=1+\frac{2}{x}[/tex]. 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 [tex]\frac{2}{x}[/tex] gets larger and larger, in a negative fashion. Then after 0, [tex]\frac{2}{x}[/tex] 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 [tex](-\inf,0] U (1,\inf)[/tex]... Then I look at when [tex]x<-2[/tex] and note that as x decreases, [tex]\frac{-2}{x}[/tex] gets closer to zero and this the function tends towards -1. So the range in this interval is [tex](-1,0}[/tex]... This gives a combined range of [tex](-\inf,0] U (1,\inf)[/tex].

Thanks a lot :)
 
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  • #4
Why is the domain obvious?
I don't get that at all.
Please tell me.
 
  • #5
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
Well, why can't its domain be [2,4], then?
Or the set of points [itex]\{-1,3.14,57\}[/itex]?

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.
 
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  • #7
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
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
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
 

Related to Computing the range for a rational function involving absolute value

1. What is a rational function involving absolute value?

A rational function involving absolute value is a mathematical expression that includes a ratio of two polynomials, with one or both of the polynomials containing an absolute value symbol.

2. How is the range of a rational function involving absolute value calculated?

The range of a rational function involving absolute value is calculated by first finding the domain of the function, and then determining the possible values for the output (or y) based on the restrictions of the absolute value expression.

3. Can a rational function involving absolute value have a finite range?

Yes, a rational function involving absolute value can have a finite range. This can occur when the absolute value expression has a limited range of possible values, resulting in a limited range for the overall function.

4. How does the value of the constant in the absolute value expression affect the range of the rational function?

The value of the constant in the absolute value expression can affect the range of the rational function by shifting the graph of the function horizontally. This can change the range by either expanding or compressing the graph, potentially widening or narrowing the range of possible output values.

5. Are there any special cases to consider when computing the range of a rational function involving absolute value?

Yes, there are two special cases to consider when computing the range of a rational function involving absolute value. The first is when the absolute value expression is equal to zero, which can result in a finite range of only one possible output value. The second is when the denominator of the rational function is equal to zero, which can result in an undefined range.

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