# Range of rational function

• FlO-rida

#### FlO-rida

how can i find the range of a rational function

for ex. y=1/x+1

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What's the definition of the range of a function?

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?

Also, what exactly is your function? Most would interpret what you have written like so:
$$y = \frac{1}{x} + 1$$

$$y = \frac{1}{x + 1}$$

When you write a fraction on a single line, use parentheses. The second version above should be written this way: y = 1/(x + 1)

ok how would we find the range for that

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.

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.

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 can't find anything on how to express the range (not in interval notation). don't get me wrong i know what range is.

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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 can't find anything on how to express the range (not in interval notation). don't 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

$$y = \frac{1}{x+1}$$

What's the domain of this function?

thats wat i mean i just didnt know how to do it

y=\\frac{1}{x+3}

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.

[x]\neq[/-1]

i just made an acc. yesterday

OK, good. So let's consider two cases:

$$x < -1$$

$$x > -1$$

Start by focusing on the first case, so we're just considering $x < -1$. For $x$ 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.)

it would go on to infinity, but wouldn't the range be expressed using y values

it would go on to infinity, but wouldn't 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 $x < -1$, then can y be positive at all?

infinitely small

so how would you express the range ({y|y...)

infinitely small

OK, so you can make y as negative as you like by varying x over the interval $(-\infty, -1)$. How large can you make y if x is in the same interval?

sorry i am still in math 20 and we didnt take interval notation yet

sorry i am still in math 20 and we didnt take interval notation yet

No problem. It's just another way of writing

$$x < -1$$

So how big can y get if x is in this interval? Can you make y be positive? Can you make y be zero?

i think you can make it zero, but can you please explain

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zero

What value of x causes y = 0?

infinity? sorry but i just don't get this

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.

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 $x < -1$, the range of possible y values is $y < 0$.

Other ways of expressing the same thing are

$$\{y | y < 0\}$$

or

$$(-\infty,0)$$

Now let's consider the other half of the domain: $x > -1$. What range of y values are possible here?

positve infinity or y > 0

positve infinity or y > 0

OK, so if $x > 1$ then the possible range of y values is $y > 0$.

Now put the two halves together to get the total range of the function.

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 $(-\infty, -1)\cup (-1, \infty)$.

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.

Thanks for the help, and just to be sure, the range is always expressed as y $$\neq$$ ...

No, only sometimes, like here.