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Offline xiankai

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inverse of a function
« on: July 27, 2006, 08:47:59 AM »
as we all know, for a function to have an inverse, it must be one-one. now, i am wondering if it applies for the inverse too.

for example,

g(x) = x2 - 1

g-1(x) = ± ? x - 1

as can be seen, the inverse is not one-one. therefore the function cannot be mapped back. yet my  calculations show the inverse to have either a positive or negative sign.

how can i remedy this?

... also, what is the relationship between a function and its inverse? i am thinking they are the reflection of each other in the line y = x, but can someone confirm this?
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Offline pantone159

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Re: inverse of a function
« Reply #1 on: July 27, 2006, 11:26:13 AM »
as can be seen, the inverse is not one-one. therefore the function cannot be mapped back. yet my  calculations show the inverse to have either a positive or negative sign.

how can i remedy this?

I think the usual 'trick' is to (arbitrarily) choose one branch of the inverse as the 'principal' one, and this branch is then one-to-one.  Without something like that, inverses of trig functions would not make much sense, for example.

... also, what is the relationship between a function and its inverse? i am thinking they are the reflection of each other in the line y = x, but can someone confirm this?

Yes, I think that is correct.  Reflecting across the line y=x is the same as swapping x and y.
Your function f defines a set of points x,f(x) and when you swap you then have the set of points f(x),x and that is the set that the inverse defines.

Offline xiankai

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Re: inverse of a function
« Reply #2 on: July 28, 2006, 08:10:50 AM »
I think the usual 'trick' is to (arbitrarily) choose one branch of the inverse as the 'principal' one, and this branch is then one-to-one.  Without something like that, inverses of trig functions would not make much sense, for example.

hmm... it sounds like i need to define a domain/range for it, is it ok?
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Offline pantone159

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Re: inverse of a function
« Reply #3 on: July 28, 2006, 01:57:53 PM »
That sounds right.

Offline xiankai

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Re: inverse of a function
« Reply #4 on: July 28, 2006, 10:09:58 PM »
very much thanks, i'll serve you a dish of my speciality, scooby a la snack! :D
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Offline FeLiXe

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Re: inverse of a function
« Reply #5 on: August 02, 2006, 09:30:44 AM »
if you are still wondering about inverse functions, that's what I heard in maths:

a function must be bijective that means injective (one-to-one) and surjective (onto) to have an inverse function. the inverse function is then also bijective

x^2-1 is not one-to-one because y=0 is reached from x=1 and x=-1

it is not onto because y=-2 is not reached at all
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Offline xiankai

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Re: inverse of a function
« Reply #6 on: August 03, 2006, 09:35:15 AM »
from what i can tell, surjective refers to all the elements in the range corresponding to one or more unique elements in the domain, am i wrong?

A --> 1
B --> 2
C --> 2
C --> 3
(surjective form)

because if so, i find it confusing that since injective refers to each element in the domain corresponding to one unique element in the range.

A --> 1
B --> 2
C --> 3
(injective form)

thus injectivity can be seen as a more specific subset of surjectivity, thus making surjectivity redundant

???
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Offline Yggdrasil

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Re: inverse of a function
« Reply #7 on: August 03, 2006, 03:47:46 PM »
Injectivity and surjectivity are only redundant if you're talking about functions whose domain has the same dimension/cardinality as its co-domain.  For example, most functions from one variable calculus map real number to real numbers.  An example of a function which maps to a set of a different cardinality is the projection function which, for example, maps elements of a two-dimensional space into a one dimensional space.  In fact, projections are good examples of functions which are surjective, but not injective.   For example, the projection onto the x-axis:

f: R2 -> R | (x,y) -> x

is surjective, but it is not injective.  An example of a function which is injective but not surjective would be a map from a one-dimensional space to a two-dimensional space:

g: R -> R2  | x -> (x,0)
« Last Edit: August 03, 2006, 04:04:00 PM by Yggdrasil »

Offline xiankai

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Re: inverse of a function
« Reply #8 on: August 04, 2006, 07:51:17 AM »
i dont know about dimensional functions but i was asking about the one-one function in particular  :-X
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Offline FeLiXe

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Re: inverse of a function
« Reply #9 on: August 04, 2006, 07:57:08 AM »
actually injectivity and surjectivity are only the same over sets of the same finite cardinality

an exponential function R->R is injective but not surjective

a typical cubic fuction R->R is surjective but not injective

the function f:N->N f(x)=2x is injective but not surjective

...
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Offline FeLiXe

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Re: inverse of a function
« Reply #10 on: August 04, 2006, 08:00:34 AM »
xiankai: maybe I should add that your g-1 is not a function because two values are assigned to each value of x

besides that I don't understand what your question is about
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Offline xiankai

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Re: inverse of a function
« Reply #11 on: August 04, 2006, 08:30:52 PM »
i guess i may be asking the wrong questions... lets start at the basics then :D

how do you define surjectivity and injectivity?
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Offline Yggdrasil

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Re: inverse of a function
« Reply #12 on: August 05, 2006, 12:22:52 AM »
Well, here are the formal definitions (plus one more needed as a basis for the others):

A function, denoted f: X -> Y, maps each element of the set X to an element in the set Y.

A function, f, is injective if f(a) = f(b) implies that a = b.

A function, f: X -> Y, is surjective if for every y in Y, there exists an x in X such that f(x) = y.

Offline FeLiXe

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Re: inverse of a function
« Reply #13 on: August 05, 2006, 09:28:54 AM »
for example
lets say we have two sets X={A,B,C} and Y={1,2,3}
and functions X->Y

A --> 1
B --> 2
C --> 2

is a function
is not injective because B and C are projected to the same value (2)
not surjective because 3 is not reached

A --> 1
B --> 2
C --> 2
C --> 3

is not a function because to one value of X two values of Y are assigned

A --> 1
B --> 3
C --> 2

is a function
is injective
is bijective
(as we said with finite sets of the same cardinality injective and surjective is the same)
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Offline xiankai

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Re: inverse of a function
« Reply #14 on: August 05, 2006, 11:05:44 AM »
hmm i think i understand it better now; im uneducated in function language but i can infer from the examples, sweet! :D
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