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Reading functions formulae

Posted on 2012-04-08
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Last Modified: 2012-06-27
Dear experts,

Can someone please help me read the below formula. The below formula is taken from functions formulae (image and preimage)

f={(x,x2+1) | x ¿R)}

I have started learning mathematics on my own, hence these questions.

Also if anyone suggest any website where i could see tutorials on this topic.

I have visited khan academy, but i have not found any tutorial on the below subject.

thank you
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Question by:Excellearner
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Author Comment

by:Excellearner
ID: 37821001
dear experts,

there has been some script issue.

It should be read x 'belongs to' R.

thankyou
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Expert Comment

by:phoffric
ID: 37821442
>> f={(x,x^2+1) | x is in R)}
Took liberty changing x2 to x-squared (is this right?)
This looks like a mapping of all x in R.
Given any point x in the x-axis (in R), then the corresponding point along the y-axis (also in R) is x-squared.
So, the function f is just a parabola: f(x) = x^2+1
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Expert Comment

by:phoffric
ID: 37821460
http://en.wikipedia.org/wiki/Image_(mathematics)
http://en.wikipedia.org/wiki/Function_(mathematics)#Image_of_a_set

(x,x^2+1) represents the set of points in R^2 defined by the function f(x) = x^2+1
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Author Comment

by:Excellearner
ID: 37821665
phorrric,

thank you fo rthe comment.

yes you are right it was x squared, i observed it much after i posted the question.

my frustation is this,

 f={(x,x^2+1) | x is in R)}

if i rewrite the above function to this, am i missing anything

 f={(x^2+1) | x is in R)}

Kindly comment.

Thank you.
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Accepted Solution

by:
phoffric earned 500 total points
ID: 37821773
This blog tries to discuss various forms of notation confusion:
    http://gowers.wordpress.com/2011/10/13/domains-codomains-ranges-images-preimages-inverse-images/

From that blog and the previous links, I am inclined to take issue with your notation:  f={(x^2+1) | x is in R)}
It seems to be infering a mapping from R -> R as x -> x^2 + 1, but I think you need to be more explicit, as in:

f: R -> R, defined by { x -> (x^2+1) | x is in R) }
or,
f: R -> R, defined by x ->  x^2 + 1 or equivalently, f(x) = x^2 + 1
   but, notationally, f(x), may be a little misleading, since, on one hand, f(x) is just a number, given a number x; yet, we have learned it in H.S. algebra that f(x) is a function. The blog tries to help clarify these points and relates algebra to set theory.

Paraphrasing from this blog and applying to your problem..
Given a function f: R -> R , we can define a natural notion of the graph of that function. It is the set of all points ( x, f(x) ) where x is an element in R. To put it another way, it is the set of all points (x, y) that are in R x R such that y = f(x) =  (x^2+1).
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Expert Comment

by:phoffric
ID: 37821776
>>  f={(x,x^2+1) | x is in R)}
So, using the graph idea noted in previous post, you can read this as f is a set of points (x, x^2+1) where x is in R, and the point is in R^2 (or, equivalently, R x R).
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