Question

Please help me understand to find the range of this function: f(x) = x^2 / (1 - x^2)

Answer 1

Hint: Find the horizontal asymptote. That will determine the range.

Answer 2

How do I do that? Would I have to first solve the function?

Answer 3

The degrees of both the numerator and denominator are the same, agreed?

Answer 4

yes

Answer 5

f(x) = x^2 / (1 - x^2)

Answer 6

since the degrees are equal, you divide the leading coefficients

Answer 7

so 1/(-1) = -1

this means that the horizontal asymptote is y = -1

Answer 8

do you see how to use this to get the range?

Answer 9

no, not really.. I am taking a precalculus course online and am having a really hard time right now understanding..

Answer 10

the horizontal asymptote is the line where the graph approaches it, but never crosses or touches it

Note: this isn't true for all cases, but it's definitely true in this case

Answer 11

so this means that you'll never be able to get an output of -1 when you plug in any x value

Answer 12

but you can get any other output you want

Answer 13

so that's why the range is the set of all real numbers y, but y can't equal -1

Answer 14

the answer says (-infinity, infinity) Union [0, infinity)

Answer 15

I am not understanding how that exactly came about

Answer 16

that's what they say what the range is?

Answer 17

if you union those two intervals, you'll get (-infinity, infinity)

Answer 18

but clearly y = -1 is not part of the range

Answer 19

yes, that is what it says..

Answer 20

can you post a screenshot of what it says?

Answer 21

#19

Answer 22

that is just bizarre...

Answer 23

ok say that the answer was (-inf, inf) U [0, inf)...it's not, but let's say it is

if you union those two intervals, you would just get (-inf, inf)

so why even bother with the [0, inf) portion?

Answer 24

but the answer can't be (-inf, inf) because -1 isn't part of the range

no matter how hard you try, you will never ever be able to get y = -1 as an output

you could try for any and all x values and it would never happen

Answer 25

do you see what I mean?

Answer 26

yes

Answer 27

so maybe there's a typo or the numbers got mixed up?

Answer 28

So if you are correct, then the range would be (-infinity, infinity) and y ≠ -1?

Answer 29

yes that's what I think the answer should be

Answer 30

essentially you can produce any output you want BUT y = -1

Answer 31

basically it would be this in interval notation

(-inf, -1) U (-1, inf)

Answer 32

Okay, well I understand how you explained it. Thank you.

Answer 33

you're welcome