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F i n d space t h e space r a n g e space o f space t h e space f u n c t i o n space f space d e f i n e d space b y space f equals fraction numerator x squared minus x plus 1 over denominator x squared plus x plus 1 end fraction comma space f o r space a l l space x greater than 0

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Asked by Topperlearning User 25th October 2016, 8:58 AM
Answered by Expert
Answer:

C o n s i d e r space t h e space g i v e n space f u n c t i o n
f open parentheses x close parentheses equals fraction numerator x squared minus x plus 1 over denominator x squared plus x plus 1 end fraction comma space f o r space a l l space x greater than 0
rightwards double arrow y equals fraction numerator x squared minus x plus 1 over denominator x squared plus x plus 1 end fraction
rightwards double arrow y open parentheses x squared plus x plus 1 close parentheses equals x squared minus x plus 1
rightwards double arrow x squared y plus x y plus y equals x squared minus x plus 1
rightwards double arrow x squared open parentheses y minus 1 close parentheses plus x open parentheses y plus 1 close parentheses plus open parentheses y minus 1 close parentheses equals 0 comma space x greater than 0 space.... left parenthesis 1 right parenthesis
S i n c e space x element of R comma space t h e space a b o v e space q u a d r a t i c space s h o u l d space h a v e space r e a l space r o o t s.
rightwards double arrow b squared minus 4 a c greater or equal than 0
rightwards double arrow open parentheses y plus 1 close parentheses squared minus 4 open parentheses y minus 1 close parentheses open parentheses y minus 1 close parentheses greater or equal than 0
rightwards double arrow negative 3 y squared plus 10 y minus 3 greater or equal than 0
rightwards double arrow 3 y squared minus 10 y plus 3 less or equal than 0
rightwards double arrow 3 y squared minus 9 y minus y plus 3 less or equal than 0
rightwards double arrow 3 y open parentheses y minus 3 close parentheses minus open parentheses y minus 3 close parentheses less or equal than 0
rightwards double arrow open parentheses 3 y minus 1 close parentheses open parentheses y minus 3 close parentheses less or equal than 0
rightwards double arrow 3 open parentheses y minus 1 third close parentheses open parentheses y minus 3 close parentheses less or equal than 0
rightwards double arrow open parentheses y minus 1 third close parentheses open parentheses y minus 3 close parentheses less or equal than 0
rightwards double arrow 1 third less or equal than y less or equal than 3... left parenthesis 2 right parenthesis

C o n s i d e r space t h e space q u a d r a t i c space left parenthesis 1 right parenthesis.
H e r e space t h e space s u m space o f space t h e space r o o t s space greater than 0 space a n d space t h e space p r o d u c t space o f space t h e space r o o t s space greater than 0 comma space sin c e space x greater than 0
rightwards double arrow negative fraction numerator open parentheses y plus 1 close parentheses over denominator open parentheses y minus 1 close parentheses end fraction greater than 0 space a n d space fraction numerator open parentheses y minus 1 close parentheses over denominator open parentheses y minus 1 close parentheses end fraction greater than 0
rightwards double arrow fraction numerator open parentheses y plus 1 close parentheses over denominator open parentheses y minus 1 close parentheses end fraction less than 0 space a n d space fraction numerator open parentheses y minus 1 close parentheses over denominator open parentheses y minus 1 close parentheses end fraction greater than 0
rightwards double arrow fraction numerator open parentheses y plus 1 close parentheses over denominator open parentheses y minus 1 close parentheses end fraction cross times space fraction numerator open parentheses y minus 1 close parentheses over denominator open parentheses y minus 1 close parentheses end fraction less than 0
rightwards double arrow fraction numerator open parentheses y squared minus 1 close parentheses over denominator open parentheses y minus 1 close parentheses squared end fraction less than 0
rightwards double arrow y squared minus 1 less than 0
rightwards double arrow y squared less than 1
rightwards double arrow y less than plus-or-minus 1
rightwards double arrow negative 1 less than y less than 1... left parenthesis 3 right parenthesis

F r o m space left parenthesis 2 right parenthesis space a n d space left parenthesis 3 right parenthesis comma space w e space f i n d space t h a t
1 third less or equal than y less than 1 space rightwards double arrow R a n g e space o f space f equals left square bracket 1 third comma 1 right parenthesis

Answered by Expert 25th October 2016, 10:58 AM
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