Question

How do you determine the range of a rational function?

Answer 1

For example, f(x) = (x+1)/(x+6)

Answer 2

The domain would be all real numbers except x = -6... what about the range?

Answer 3

The range is all possibly output values. The range of function is the domain of the inverse.
\[y=(6x-1)/(1-x)\]
The domain of this inverse function is such that x cannot be 1, therefore the range of the original is that y cannot equal 1.

Answer 4

possible*

Answer 5

There's ways but they aren't very nice.

You can find the inverse of the function (not always possible and generally not easy), and then find the domain of that (which is the range of the original function).

You can also look at the graphs behaviour near the vertical asymptotes (whether it goes toward positive or negative infinity near it).

Or there's calculus.

Answer 6

You lost me... to find the range, I first find the inverse of the rational function?

Answer 7

Okay

Answer 8

Yes, find the inverse of the original function.

Answer 9

I'm only in precalc, so I'll save the calculus for next year ;)

Answer 10

The inverse method will only work with pretty simple rational functions, though. Graphs of rational functions are always helpful for finding range.

Answer 11

This is precalc, I doubt the functions given are that complicated.

Answer 12

Okay, for the inverse I got...
y( = (-6x+1)/(x-1)

Does that look right?

Answer 13

Yep, same as Cosmo's.

Answer 14

Would finding the asymptotes help with the range? we've been learning about that in previous lessons

Answer 15

Yes those can too. Horiz. ones.

Answer 16

Refresh my memory... how do you find horizontal asymptotes again?

Answer 17

They're just the values that are excluded from the domain or range. It's the same thing.

Answer 18

Would x = 1 be a horizontal asymptote then?

Answer 19

Horiz asymptotes - look at the degree of numerator and denominator.

Answer 20

Same degree, 1

Answer 21

Then you divide the coefficients, in this case both are 1 so y=1.

Answer 22

But keep in mind some rational functions can cross the horiz. asymptote (they're only end-behaviour asymptotes).

Answer 23

So horiz. asymptotes wouldn't really help with finding the range then? since functions can cross them anyway? thanks for the help!

Answer 24

It can but won't necessarily. If you find the asymptote, you can look at the graph to check it doesn't cross it.

The example you gave - simple ones like that never cross the asymptotes.

Answer 25

So for my precalc problems, to find the ranges I should stick to finding the inverse functions? and also maybe looking at the horiz. asymptotes?

Answer 26

Yeah, the ones you're asked to find the range of, will likely be the ones you can find inverses of. You shouldn't be asked for the range of something like http://zonalandeducation.com/mmts/functionInstitute/rationalFunctions/definition/rfdef2.gif

Answer 27

Hmm actually that looks like some of the problems I'm being given..

Answer 28

But the graph of that makes it pretty clear the range is all real numbers