CBSE Class 12-science Answered

$L e t space f colon N rightwards arrow N space b e space d e f i n e d space b y space f open parentheses n close parentheses equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator n plus 1 over denominator 2 end fraction comma space i f space n space i s space o d d end cell row cell n over 2 comma space i f space n space i s space e v e n end cell end table close comma space f o r space a l l space n element of N S t a t e space w h e t h e r space t h e space f u n c t i o n space f space i s space b i j e c t i v e. space J u s t i f y space y o u r space a n s w e r.$ Please provide solution to this question

Asked by Topperlearning User | 25 Oct, 2016, 08:59: AM

Expert Answer

$G i v e n space f colon N rightwards arrow N space b e space d e f i n e d space b y space f open parentheses n close parentheses equals open curly brackets table attributes columnalign left end attributes row cell fraction numerator n plus 1 over denominator 2 end fraction comma space i f space n space i s space o d d end cell row cell n over 2 comma space i f space n space i s space e v e n end cell end table close comma space f o r space a l l space n element of N W e space n o t e space t h a t space f open parentheses 1 close parentheses equals fraction numerator 1 plus 1 over denominator 2 end fraction equals 1 comma space f open parentheses 2 close parentheses equals 2 over 2 equals 1 f open parentheses 3 close parentheses equals fraction numerator 3 plus 1 over denominator 2 end fraction equals 2 comma space f open parentheses 4 close parentheses equals 4 over 2 equals 2 I n space g e n e r a l comma f open parentheses 2 m minus 1 close parentheses equals fraction numerator open parentheses 2 m minus 1 close parentheses plus 1 over denominator 2 end fraction equals m space a n d space f open parentheses 2 m close parentheses equals fraction numerator 2 m over denominator 2 end fraction equals m rightwards double arrow f open parentheses 2 m minus 1 close parentheses equals f open parentheses 2 m close parentheses comma space w h e r e space m space i s space a n y space p o s i t i v e space i n t e g e r. rightwards double arrow f space i s space n o t space o n e minus o n e F o r space a n y space n element of N comma space t h e r e space i s space 2 n element of N comma space s u c h space t h a t space f open parentheses 2 n close parentheses equals fraction numerator 2 n over denominator 2 end fraction equals n T h u s comma space f space i s space o n t o space b u t space n o t space o n e minus o n e. space T h e r e f o r e comma space f space i s space n o t space a space b i j e c t i o n.$