What is the domain of f(x)= 2/(sin(3x))+sin(x) ?

The domain of f(x)= 2/(sin(3x))+sin(x) is

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domain of f(x)= 2/(sin(3x))+sin(x)

domain

f (x) = \frac{2}{sin ( 3 x )} + sin (x)

2 πn < x < \frac{π}{3} + 2 πn or \frac{π}{3} + 2 πn < x < \frac{2 π}{3} + 2 πn or \frac{2 π}{3} + 2 πn < x < π + 2 πn or π + 2 πn < x < \frac{4 π}{3} + 2 πn or \frac{4 π}{3} + 2 πn < x < \frac{5 π}{3} + 2 πn or \frac{5 π}{3} + 2 πn < x < 2 π + 2 πn

Interval Notation

(2 πn, \frac{π}{3} + 2 πn) \cup (\frac{π}{3} + 2 πn, \frac{2 π}{3} + 2 πn) \cup (\frac{2 π}{3} + 2 πn, π + 2 πn) \cup (π + 2 πn, \frac{4 π}{3} + 2 πn) \cup (\frac{4 π}{3} + 2 πn, \frac{5 π}{3} + 2 πn) \cup (\frac{5 π}{3} + 2 πn, 2 π + 2 πn)

Solution steps

Take the denominator(s) of

\frac{2}{sin ( 3 x )} + sin (x)

and compare to zero:

x = 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = \frac{π}{3} + 2 πn, x = π + 2 πn, x = \frac{5 π}{3} + 2 πn

Periodicity of

\frac{2}{sin ( 3 x )} + sin (x) : 2 π

Combine period:

2 πn \leq x < 2 π + 2 πn

and undefined points.

The domain is

2 πn < x < \frac{π}{3} + 2 πn or \frac{π}{3} + 2 πn < x < \frac{2 π}{3} + 2 πn or \frac{2 π}{3} + 2 πn < x < π + 2 πn or π + 2 πn < x < \frac{4 π}{3} + 2 πn or \frac{4 π}{3} + 2 πn < x < \frac{5 π}{3} + 2 πn or \frac{5 π}{3} + 2 πn < x < 2 π + 2 πn

arctan (\frac{1}{x})

χ^{3} - 4 χ

f (x) = sin (ln (x))

f (x) = \frac{6 x - 1}{x ^{2} + 1}

4 x^{2}

What is the domain of f(x)= 2/(sin(3x))+sin(x) ?
The domain of f(x)= 2/(sin(3x))+sin(x) is