Unformatted text preview: g ◦ f is nonincreasing and therefore monotone. 4.10. By a previous exercise, the equations f ( x ) = y and g ( x ) = y each have unique solutions for every y ∈ R . This says precisely that f and g are bijective. Now g ( f ( x )) = c ( ax + b ) + d = acx + bc + d and f ( g ( x )) = a ( cx + d ) + b = acx + ad + b , so g ◦ ff ◦ g is a constant function and hence is neither injective nor surjective. 4.11. Multiplication by 2 is both surjective and injective on R since 2( x/ 2) = x and if 2 x = 2 y , then x = y . Thus it is a bijection on R . It is not surjective on Z , since the image of any element of Z under multiplication by 2 is even. Therefore multiplication by 2 is not bijective on Z . (Note that it remains injective on Z since Z ⊂ R .) 1...
View
Full Document
 Spring '10
 Math, Basic concepts in set theory, Types of functions, r., Functions and mappings

Click to edit the document details