Maths-
General
Easy

Question

The domain of the function  f left parenthesis x right parenthesis equals fraction numerator tan invisible function application 2 x over denominator 6 cos invisible function application x plus 2 sin invisible function application 2 x end fraction text  is  end text

  1. R minus open curly brackets left parenthesis 2 n plus 1 right parenthesis pi over 2 colon n element of Z close curly brackets
  2. R minus open curly brackets left parenthesis 2 n plus 1 right parenthesis pi over 4 colon n element of Z close curly brackets
  3. R minus open curly brackets left parenthesis 2 n plus 1 right parenthesis pi over 2 colon n element of Z close curly brackets
  4. R minus open curly brackets left parenthesis 2 n plus 1 right parenthesis pi over 4 colon n element of Z close curly brackets

Hint:

in this question, we have to find the domain of f(x). First considering the numerator and finding its domain and later considering the denominator and finding its domain. Further, find the total domains that satisfy the equation.

The correct answer is: R minus open curly brackets left parenthesis 2 n plus 1 right parenthesis pi over 2 colon n element of Z close curly brackets


    f left parenthesis x right parenthesis equals fraction numerator tan space 2 x over denominator 6 cos space x plus 2 space sin space 2 x space end fraction
A s space tan space i s space u n d e f i n e d space a t space o d d space i n t e g r a l space m u l t i p l e space o f straight pi over 2.
2 x space not equal to space straight pi over 2 left parenthesis 2 straight n plus 1 right parenthesis
straight x space space not equal to fraction numerator straight pi left parenthesis 2 straight n plus 1 right parenthesis over denominator 4 end fraction

Now comma space for space denominator comma space
6 cos space straight x space plus 2 space sin 2 straight x space not equal to 0
6 space cos space straight x space plus 2.2 space sin space straight x space cos space straight x space not equal to 0
2 space cos space straight x left parenthesis 3 plus 2 sin space straight x right parenthesis not equal to 0
2 space cos space straight x space not equal to 0 space space space space 0 straight r space 3 plus 2 sin space straight x space not equal to 0
For comma 2 space cos space straight x space not equal to 0 rightwards double arrow cos space straight x not equal to 0
For comma space 3 plus 2 sin space straight x not equal to 0 rightwards double arrow 2 sin space straight x not equal to negative 3 rightwards double arrow sinx space not equal to fraction numerator negative 3 over denominator 2 end fraction
minimum space value space of space sin space straight x space is space minus 1 space.
cos space not equal to 0 comma space cos space is space 0 space at space odd space integral space multiple space of space straight pi over 2.
so comma space straight x not equal to fraction numerator left parenthesis 2 straight n plus 1 right parenthesis straight pi over denominator 2 end fraction
so comma d o m a i n space o f space f left parenthesis x right parenthesis space i s space R minus open curly brackets left parenthesis 2 n plus 1 right parenthesis pi over 2 colon n element of Z close curly brackets

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