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clooneyisagenius

Member

Registered: 2007-03-25

Posts: 56

injective/surjective/bijective help!!!!

True or False? Counterexample or Proof?
a) Every decreasing function from R to R is surjective
b) Every nondecreasing function from R to R is injective
c) Every injective function from R to R is monotone
d) Every surjective function from R to R is unbounded
e) Every unbounded function from R to R is surjective
f) f(x) = ax + b is both surjective and injective

My thoughts (and HELP on areas of WHY they are or are not).
a. true, HELP.
b. false, f(x)=x^2 for example.
c. no idea, HELP.
d. true, HELP
e. true, HELP
f) true, HELP.

Thanks so much.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: injective/surjective/bijective help!!!!

Do functions have to be continuous?  If no, then...

a) Every decreasing function from R to R is surjective

False.  f(x) = -x for all x <= 0, f(x) = -x - 1 for all x > 0.

b) Correct.

c) False.  f(x) = -x for all x <= 0, f(x) = x - 1 for all 0 < x < 1, f(x) = -x - 2 for all x >= 1.

d) Every surjective function from R to R is unbounded

True.  Let f(x) be a surjective function that is bounded by M.  By the Archimedean Principle, there exists a b in R such that y > M.  Thus, there exists an a in R such that f(a) = b > M.  Contradiction.

e) Every unbounded function from R to R is surjective

False.  Just cause a function goes to infinity does not mean it goes to negative infinity.

f) f(x) = ax + b is both surjective and injective

True.  Just show 1-1 and onto, quite simple to do.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: injective/surjective/bijective help!!!!

I think (b) is false:

Nondecreasing is not the same as increasing.

And (c) would be true if f is continuous.