Chemical Forums
Specialty Chemistry Forums => Other Sciences Question Forum => Topic started by: xiankai on July 27, 2006, 08:47:59 AM
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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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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.
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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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That sounds right.
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very much thanks, i'll serve you a dish of my speciality, scooby a la snack! :D
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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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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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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)
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i dont know about dimensional functions but i was asking about the one-one function in particular :-X
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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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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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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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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.
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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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hmm i think i understand it better now; im uneducated in function language but i can infer from the examples, sweet! :D