What is the domain of f(x)=-5csc((3pi)/4 x-pi/3)+1 ?

The domain of f(x)=-5csc((3pi)/4 x-pi/3)+1 is

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domain of f(x)=-5csc((3pi)/4 x-pi/3)+1

domain

\frac{8}{3} n \leq x < \frac{4}{9} + \frac{8}{3} n or \frac{4}{9} + \frac{8}{3} n < x < \frac{4}{3} + \frac{4}{9} + \frac{8}{3} n or \frac{4}{3} + \frac{4}{9} + \frac{8}{3} n < x < \frac{8}{3} + \frac{8}{3} n

Interval Notation

[\frac{8}{3} n, \frac{4}{9} + \frac{8}{3} n) \cup (\frac{4}{9} + \frac{8}{3} n, \frac{4}{3} + \frac{4}{9} + \frac{8}{3} n) \cup (\frac{4}{3} + \frac{4}{9} + \frac{8}{3} n, \frac{8}{3} + \frac{8}{3} n)

Solution steps

Take the denominator(s) of

- 5 csc (\frac{3 π}{4} x - \frac{π}{3}) + 1

and compare to zero:

x = \frac{4}{9} + \frac{8}{3} n, x = \frac{4}{3} + \frac{4}{9} + \frac{8}{3} n

Periodicity of

- 5 csc (\frac{3 π}{4} x - \frac{π}{3}) + 1 : \frac{8}{3}

Combine period:

\frac{8}{3} n \leq x < \frac{8}{3} + \frac{8}{3} n

and undefined points.

The domain is

\frac{8}{3} n \leq x < \frac{4}{9} + \frac{8}{3} n or \frac{4}{9} + \frac{8}{3} n < x < \frac{4}{3} + \frac{4}{9} + \frac{8}{3} n or \frac{4}{3} + \frac{4}{9} + \frac{8}{3} n < x < \frac{8}{3} + \frac{8}{3} n

What is the domain of f(x)=-5csc((3pi)/4 x-pi/3)+1 ?
The domain of f(x)=-5csc((3pi)/4 x-pi/3)+1 is