Range of rational function

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  • #1
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how can i find the range of a rational function

for ex. y=1/x+1
 
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  • #2
jbunniii
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What's the definition of the range of a function?
 
  • #3
Think what "domain" and "range" are. What can "x" be, and what can "x" not be? With that in mind, what can "y" be and what can "y" not be?
 
  • #4
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Also, what exactly is your function? Most would interpret what you have written like so:
[tex]y = \frac{1}{x} + 1[/tex]

I suspect that what you really meant was this:
[tex]y = \frac{1}{x + 1} [/tex]

When you write a fraction on a single line, use parentheses. The second version above should be written this way: y = 1/(x + 1)
 
  • #5
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ok how would we find the range for that
 
  • #6
Do you know what domain and range are? Your instructor would not be giving you rational functions without a thorough treatment of the concepts of domain and range, and how they relate to rational functions.
 
  • #7
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ok how would we find the range for that
The range for what? As I already said in post 4, it's not clear what you're working with.
 
  • #8
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the range for 1/(x+1) .the thing is that i am doing an assignment on rational functions that is ment to be completed without help from the teacher (it will be explained upon completion). i have already researched horizontal and vertical asymptote of rational functions, as well as the domain but i still cant find anything on how to express the range (not in interval notation). dont get me wrong i know what range is.
 
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  • #9
jbunniii
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the range for 1/(x+1) .the thing is that i am doing an assignment on rational functions that is ment to be completed without help from the teacher (it will be explained upon completion). i have already researched horizontal and vertical asymptote of rational functions, as well as the domain but i still cant find anything on how to express the range (not in interval notation). dont get me wrong i know what range is.

[EDIT] Weird, it was all garbled on my screen when I responded, but now it looks fine, both in my quoted version and in the original post!

OK, so the function of interest is

[tex]y = \frac{1}{x+1}[/tex]

What's the domain of this function?
 
  • #10
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thats wat i mean i just didnt know how to do it
 
  • #11
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y=\\frac{1}{x+3}
 
  • #12
jbunniii
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You can click on my equation (or any typeset equation on this site) and it will give you a pop-up window with the Latex code that produced it.
 
  • #13
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[x]\neq[/-1]
 
  • #14
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i just made an acc. yesterday
 
  • #15
jbunniii
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OK, good. So let's consider two cases:

[tex]x < -1[/tex]

[tex]x > -1[/tex]

Start by focusing on the first case, so we're just considering [itex]x < -1[/itex]. For [itex]x[/itex] in this range, can I make the function as big as I like? Can I make it as small as I like? If not, then what are some bounds? (Even if they're not the tightest possible bounds, it's a start.)
 
  • #16
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it would go on to infinity, but wouldnt the range be expressed using y values
 
  • #17
jbunniii
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it would go on to infinity, but wouldnt the range be expressed using y values

Yes, the range is expressed using y values.

What do you mean by "it would go on to infinity"? Can you make it infinitely large (positive)? Can you make it infinitely small (negative)?

If [itex]x < -1[/itex], then can y be positive at all?
 
  • #18
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infinitely small
 
  • #19
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so how would you express the range ({y|y...)
 
  • #20
jbunniii
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infinitely small

OK, so you can make y as negative as you like by varying x over the interval [itex](-\infty, -1)[/itex]. How large can you make y if x is in the same interval?
 
  • #21
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sorry i am still in math 20 and we didnt take interval notation yet
 
  • #22
jbunniii
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sorry i am still in math 20 and we didnt take interval notation yet

No problem. It's just another way of writing

[tex]x < -1[/tex]

So how big can y get if x is in this interval? Can you make y be positive? Can you make y be zero?
 
  • #23
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i think you can make it zero, but can you please explain
 
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  • #25
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infinity? sorry but i just dont get this
 
  • #26
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i got that from punching in random x values on my graphing calc. (10, 20, 30) and as they got bigger the y values got smaller but never reached zero.
 
  • #27
jbunniii
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i got that from punching in random x values on my graphing calc. (10, 20, 30) and as they got bigger the y values got smaller but never reached zero.

That's right. You can make y as close to 0 as you like, but it never quite gets there. You can also make y as negative as you like.

So that means that for [itex]x < -1[/itex], the range of possible y values is [itex]y < 0[/itex].

Other ways of expressing the same thing are

[tex]\{y | y < 0\}[/tex]

or

[tex](-\infty,0)[/tex]

Now let's consider the other half of the domain: [itex]x > -1[/itex]. What range of y values are possible here?
 
  • #28
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positve infinity or y > 0
 
  • #29
jbunniii
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positve infinity or y > 0

OK, so if [itex]x > 1[/itex] then the possible range of y values is [itex]y > 0[/itex].

Now put the two halves together to get the total range of the function.
 
  • #30
HallsofIvy
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The "natural domain" of a function where you are given a formula for it is just the set of all x values to which the formula can be applied. Here, the only arithmetic operation is "divide by x+ 1". You can divide any number except 0 so you can do that calculation for any number except x= -1. The domain is "all x except -1".

In interval notation that would be [itex](-\infty, -1)\cup (-1, \infty)[/itex].

The "range" is the set of possible y values.

One way of finding the range is to try to invert the function. If y= 1/(x+1) then x+1= 1/y and x= (1/y)- 1. Since we can divide by any number except 0, y can take any value except 0. The range is "all y except 0".

For this simple function, you could also have thought, since a/b= 0 gives immediately a= 0 by multiplying both sides by b, "a fraction is 0 if and only if its numerator is 0". Here the numerator is the constant 1 which is never 0. y can never be 0.
 

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