Find the range of  f(x) = 1/(1-2cosx)

Asked by ahuja8087 | 10th Sep, 2015, 04:15: PM

Expert Answer:

R a n g e space i s space t h e space s e t space o f space a l l space v a l u e s space t h a t space apostrophe y apostrophe space c a n space t a k e. y equals f open parentheses x close parentheses equals fraction numerator 1 over denominator 1 minus 2 cos x end fraction H e r e space y space c a n space a s s u m e space a l l space t h e space v a l u e s space e x c e p t space w h e n space 1 minus 2 cos x equals 0 rightwards double arrow 1 equals 2 cos x rightwards double arrow 1 half equals cos x rightwards double arrow cos pi over 3 equals cos x rightwards double arrow x equals straight pi over 3 C o n s i d e r space t h e space i n t e r v a l space open parentheses 0 comma 2 straight pi close parentheses S o space i n t h e space a b o v e space i n t e r v a l comma space x space c a n n o t space t a k e space straight pi over 3 space a n d space fraction numerator 5 straight pi over denominator 3 end fraction T h u s comma space t h e space D o m a i n space i s space s e t space o f space a l l space x comma space s u c h space t h a t x element of left square bracket 0 comma straight pi over 3 right parenthesis union open parentheses straight pi over 3 comma fraction numerator 5 straight pi over denominator 3 end fraction close parentheses union left parenthesis fraction numerator 5 straight pi over denominator 3 end fraction comma 2 straight pi right square bracket When space straight x equals 0 comma space we space have comma space straight y equals fraction numerator 1 over denominator 1 minus 2 cos 0 end fraction equals fraction numerator 1 over denominator 1 minus 2 end fraction equals fraction numerator negative 1 over denominator 2 end fraction rightwards double arrow straight f open parentheses straight x close parentheses less or equal than negative 1 Thus comma space maximum space range space for space the space interval comma space left square bracket 0 comma straight pi over 3 right parenthesis space is space minus 1 When space straight x equals straight pi comma space we space have comma space space straight y equals fraction numerator 1 over denominator 1 minus 2 cosπ end fraction equals fraction numerator 1 over denominator 1 plus 2 end fraction equals 1 third Thus comma space minimum space range space for space the space interval comma space open parentheses straight pi over 3 comma fraction numerator 5 straight pi over denominator 3 end fraction close parentheses space is space 1 third Again space range space for space the space interval space left parenthesis fraction numerator 5 straight pi over denominator 3 end fraction comma 2 straight pi right square bracket space is space minus 1.  Therefore comma space the space range space forthe space function space is space straight y element of straight R comma space and space straight y less or equal than negative 1 space and space straight y greater or equal than 1 third

Answered by Vimala Ramamurthy | 11th Sep, 2015, 09:17: AM