Find range

Asked by appiniakhil | 7th Apr, 2020, 07:43: PM

Expert Answer:

TO find the range of the function f(x)=1/sqrt{[cos x] - [sin x]}
negative 1 less or equal than cos space x less or equal than 1 comma space space f o r space a l l space x element of R
minus 1 less or equal than sin space x less or equal than 1 comma space space f o r space a l l space x element of R
T h e r e space a r e space o n l y space 3 space v a l u e s space p o s s i b l e space e a c h space f o r space left square bracket cos space x right square bracket space a n d space left square bracket sin space x right square bracket
W h e n space minus 1 less or equal than cos space x less than 0 space rightwards double arrow space left square bracket cos space x right square bracket equals negative 1 comma space 0 less or equal than cos space x less than 1 rightwards double arrow space left square bracket cos space x right square bracket equals 0 comma space c os space x equals 1 space rightwards double arrow left square bracket cos space x right square bracket equals 1
W h e n space minus 1 less or equal than sin space x less than 0 space rightwards double arrow space left square bracket sin space x right square bracket equals negative 1 comma space 0 less or equal than sin space x less than 1 rightwards double arrow space left square bracket sin space x right square bracket equals 0 comma space sin space x equals 1 space rightwards double arrow left square bracket sin space x right square bracket equals 1
F i x i n g space left square bracket cos space x right square bracket comma space space left square bracket cos space x right square bracket equals negative 1 comma space
f left parenthesis x right parenthesis equals fraction numerator 1 over denominator square root of negative 1 space minus space left square bracket sin space x right square bracket end root end fraction equals fraction numerator 1 over denominator square root of negative 1 space minus space left parenthesis negative 1 right parenthesis end root end fraction space o r space fraction numerator 1 over denominator square root of negative 1 space minus space left parenthesis 0 right parenthesis end root end fraction space space o r space space fraction numerator 1 over denominator square root of negative 1 space minus space left parenthesis 1 right parenthesis end root end fraction
rightwards double arrow f left parenthesis x right parenthesis equals 1 over 0 space comma space fraction numerator 1 over denominator square root of negative 1 end root end fraction comma space space fraction numerator 1 over denominator square root of negative 2 end root end fraction space space space
f left parenthesis x right parenthesis space i s space n o t space d e f i n e d space f o r space a l l space t h r e e
W h e n space space left square bracket cos space x right square bracket equals 0 comma space
f left parenthesis x right parenthesis equals fraction numerator 1 over denominator square root of space 0 minus space left square bracket sin space x right square bracket end root end fraction equals fraction numerator 1 over denominator square root of 0 minus space left parenthesis negative 1 right parenthesis end root end fraction space o r space fraction numerator 1 over denominator square root of 0 space minus space left parenthesis 0 right parenthesis end root end fraction space space o r space space fraction numerator 1 over denominator square root of 0 space minus space left parenthesis 1 right parenthesis end root end fraction
rightwards double arrow f left parenthesis x right parenthesis equals 1 over 1 space comma space fraction numerator 1 over denominator square root of 0 end fraction comma space space fraction numerator 1 over denominator square root of negative 1 end root end fraction space space space
fraction numerator 1 over denominator square root of 0 end fraction comma space space fraction numerator 1 over denominator square root of negative 1 end root end fraction space a r e space n o t space d e f i n e d space
rightwards double arrow f left parenthesis x right parenthesis equals 1
W h e n space space left square bracket cos space x right square bracket equals 1 comma space
f left parenthesis x right parenthesis equals fraction numerator 1 over denominator square root of space 1 minus space left square bracket sin space x right square bracket end root end fraction equals fraction numerator 1 over denominator square root of 1 minus space left parenthesis negative 1 right parenthesis end root end fraction space o r space fraction numerator 1 over denominator square root of 1 space minus space left parenthesis 0 right parenthesis end root end fraction space space o r space space fraction numerator 1 over denominator square root of 1 space minus space left parenthesis 1 right parenthesis end root end fraction
rightwards double arrow f left parenthesis x right parenthesis equals fraction numerator 1 over denominator square root of 2 end fraction space comma space 1 over 1 comma space space fraction numerator 1 over denominator square root of 0 end fraction space space space
fraction numerator 1 over denominator square root of 0 end fraction space space i s space n o t space d e f i n e d space
f left parenthesis x right parenthesis equals open curly brackets 1 comma fraction numerator 1 over denominator square root of 2 end fraction close curly brackets
H e n c e comma space t h e space r a n g e space o f space f left parenthesis x right parenthesis space i s space open curly brackets 1 comma fraction numerator 1 over denominator square root of 2 end fraction close curly brackets.

Answered by Renu Varma | 8th Apr, 2020, 11:09: AM