hw5solutions23b

hw5solutions23b - Anna Medvedovsky [email protected] Math...

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Anna Medvedovsky [email protected] Math 23b / Spring 2009 HW #3 solutions 1. For polynomial functions f ( x ) = x - 1 and g ( x ) = x 2 - 1 find f g and g f . Computing, f g ( x ) = x 2 - 2 and g f ( x ) = x 2 - 2 x. 2. Explain why multiplication by 2 defines a bijection from R to R but not from Z to Z . Multiplication by 2 is not a surjective function from Z to Z . For example, there’s no element of Z that you can multiply by 2 to get 1. So multiplication by 2 is not bijective from Z to Z . But multiplication by 2 is bijective from R to R because dividing by 2 defines an inverse function. (Note that dividing by 2 does not define a function from Z to Z .) 3. Let f : R R be a bijective increasing function. Prove that f - 1 : R R is an increasing function as well. Given x > y in R , we want to show that f - 1 ( x ) > f - 1 ( y ). So suppose not. Then f - 1 ( x ) f - 1 ( y ), so that either f - 1 ( x ) = f - 1 ( y ) or f - 1 ( x ) < f - 1 ( y ). If f - 1 ( x ) = f - 1 ( y ), then x = f ( f - 1 ( x ) ) = f ( f - 1 ( y ) ) = y. And if f - 1 ( x ) < f - 1 ( y ), then, since f is an increasing function, we must have x = f ( f - 1 ( x ) ) < f ( f - 1 ( y ) ) = y. In either case,
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This note was uploaded on 10/15/2011 for the course MATH 23b taught by Professor Bonglian during the Spring '09 term at Brandeis.

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hw5solutions23b - Anna Medvedovsky [email protected] Math...

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