Range of rational function

  • Thread starter FlO-rida
  • Start date
  • #26
26
0
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
Science Advisor
Homework Helper
Insights Author
Gold Member
3,473
255
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
26
0
positve infinity or y > 0
 
  • #29
jbunniii
Science Advisor
Homework Helper
Insights Author
Gold Member
3,473
255
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
Science Advisor
Homework Helper
41,833
964
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.
 
  • #31
26
0
Thanks for the help, and just to be sure, the range is always expressed as y [tex]\neq[/tex] ....
 
  • #32
Char. Limit
Gold Member
1,204
14
No, only sometimes, like here.
 

Related Threads on Range of rational function

  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
8
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
10
Views
3K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
17
Views
750
  • Last Post
Replies
4
Views
920
  • Last Post
Replies
8
Views
2K
Replies
1
Views
662
Top