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Needs help in function

Define a function f: R-->R by the formula f(x) = 3x - 5.
a. Prove that f is one-to-one
b. Prove that f is onto.

Thanks.
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mustish1
Asked:
mustish1
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2 Solutions
 
TommySzalapskiCommented:
Prove it by contradiction. Assume that it is not one-to-one, then find a contradiction.
If it is not one-to-one, then there exist two values x1 and x2 such that f(x1) = f(x2) but x1 not= x2 or x1 = x2 and f(x1) not= f(x2)
Continue this until it fails miserably. The onto proof is very similar.

This is an academic question so to facilitate learning, please no one just jump in with a full solution.
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GwynforWebCommented:
Keep it simple is the key to this question.

Suppose y= 3x-5 then

   x= (y+5)/3

then for

(1) For any y in R there is only one possible x hence 1-1.  ( given by x= (y+5)/3 )

(2) For any y in R there is an x such y =  3x-5.  ( given by x= (y+5)/3 )
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TommySzalapskiCommented:
To me that doesn't seem like the kind of rigorous proof that would be required for that type of question. If you do have rules that you can use though to show one-to-one, then use them of course.
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TommySzalapskiCommented:
The goal, of course, isn't to prove one-to-one and onto; it's to prove them given the constructs that have been provided to you. Since we have no idea what those are, we can only throw out suggestions. Gwen's solution is very intuative and anyone can understand it fairly easily. If it works for your class/self-learing/whatever, then it's much simpler than mine and it great. If you are in some kind of foundations or logic class where you need to use very specific rules, then proof by contradiction is almost always included.
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GwynforWebCommented:
... my guess is that is an introductory course on functions. I doubt proof by contradiction has been covered yet.
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TommySzalapskiCommented:
Could be.
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