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what is the domain of f(x) = cos^-1 (2x/1+x)

Asked by pdcavita 25th April 2019, 10:11 PM
Answered by Expert
Answer:
G i v e n colon space f left parenthesis x right parenthesis equals cos to the power of negative 1 end exponent open parentheses fraction numerator 2 x over denominator 1 plus x end fraction close parentheses
T o space f i n d space t h e space d o m a i n space o f space f left parenthesis x right parenthesis
cos space i n v e r s e space i s space d e f i n e d space f o r space left square bracket negative 1 comma 1 right square bracket space sin c e space cos space t a k e s space v a l u e s space f r o m space minus 1 space t o space 1
rightwards double arrow fraction numerator 2 x over denominator 1 plus x end fraction element of left square bracket negative 1 comma 1 right square bracket
rightwards double arrow negative 1 less or equal than fraction numerator 2 x over denominator 1 plus x end fraction less or equal than 1
C o n s i d e r space L e f t space s i d e space i n e q u a l i t y space f i r s t space a n d space t h e n space s o l v e space i t
minus 1 less or equal than fraction numerator 2 x over denominator 1 plus x end fraction
rightwards double arrow 0 less or equal than 1 plus fraction numerator 2 x over denominator 1 plus x end fraction
rightwards double arrow fraction numerator 1 plus x plus 2 x over denominator 1 plus x end fraction greater or equal than 0
rightwards double arrow fraction numerator 1 plus 3 x over denominator 1 plus x end fraction greater or equal than 0
F o r space t h e space a b o v e space i n e q a l i t y space t r u e comma space b o t h space n u m e r a t o r space a n d space d e n o m i n a t o r space h a s space t o space b e space e i t h e r space p o s i t i v e space o r space n e g a t i v e space a t space t h e space s a m e space t i m e
rightwards double arrow 1 plus 3 x greater or equal than 0 space & space 1 plus x greater or equal than 0 space space space O R space space thin space 1 plus 3 x less or equal than 0 space & space 1 plus x less or equal than 0
rightwards double arrow x greater or equal than negative 1 third space & space x greater or equal than negative 1 space space space O R space space thin space x less or equal than negative 1 third space & space x less or equal than negative 1
rightwards double arrow x greater or equal than negative 1 third space space space O R space space space x less or equal than negative 1
rightwards double arrow x element of left square bracket negative 1 third comma infinity right parenthesis space union space left parenthesis negative infinity comma negative 1 right parenthesis
rightwards double arrow x element of space left parenthesis negative infinity comma negative 1 right parenthesis space union space left square bracket negative 1 third comma infinity right parenthesis space.......... space left parenthesis i right parenthesis
T a k i n g space t h e space r i g h t space s i d e space i n e q u a l i t y
fraction numerator 2 x over denominator 1 plus x end fraction less or equal than 1
rightwards double arrow fraction numerator 2 x over denominator 1 plus x end fraction minus 1 less or equal than 0
rightwards double arrow fraction numerator x minus 1 over denominator 1 plus x end fraction less or equal than 0
T h e space a b o v e space i n e q u a l i t y space i s space t r u e space i f space e i t h e r space o f space t h e space b o t h space i s space p o s i t i v e space a n d space o t h e r space i s space n e g a t i v e
rightwards double arrow x minus 1 greater or equal than 0 space & space 1 plus x less or equal than 0 space O R space x minus 1 less or equal than 0 space & space 1 plus x greater or equal than 0
rightwards double arrow x greater or equal than 1 space & space x less or equal than negative 1 space O R space x less or equal than 1 space & space x greater or equal than negative 1
rightwards double arrow 1 less or equal than x less or equal than negative 1 space space O R space space space minus 1 less or equal than x less or equal than 1
T h e space f i r s t space c a s e space i s space n o t space p o s s i b l e space a n d space t h u s space w e space h a v e
minus 1 less or equal than x less or equal than 1 space rightwards double arrow space x element of left square bracket negative 1 comma 1 right square bracket space......... space left parenthesis i i right parenthesis
F r o m space left parenthesis i right parenthesis space a n d space left parenthesis i i right parenthesis
x element of space left parenthesis negative infinity comma negative 1 right parenthesis space union space left square bracket negative 1 third comma infinity right parenthesis space intersection space left square bracket negative 1 comma 1 right square bracket
rightwards double arrow x element of space left square bracket negative 1 third comma 1 right square bracket
H e n c e comma space t h e space d o m a i n space o f space f left parenthesis x right parenthesis space i s space t h e space i n t e r v a l space left square bracket negative 1 third comma 1 right square bracket.
Answered by Expert 26th April 2019, 11:15 AM
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