What is the domain of 2csc(1/3 (x-1))-2 ?

The domain of 2csc(1/3 (x-1))-2 is 6pin<= x<1+6pin\lor 1+6pin<x<3pi+1+6pin\lor 3pi+1+6pin<x<6pi+6pin

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domain of 2csc(1/3 (x-1))-2

domain

2 csc (\frac{1}{3} (x - 1)) - 2

6 πn \leq x < 1 + 6 πn or 1 + 6 πn < x < 3 π + 1 + 6 πn or 3 π + 1 + 6 πn < x < 6 π + 6 πn

Interval Notation

[6 πn, 1 + 6 πn) \cup (1 + 6 πn, 3 π + 1 + 6 πn) \cup (3 π + 1 + 6 πn, 6 π + 6 πn)

Solution steps

Take the denominator(s) of

2 csc (\frac{1}{3} (x - 1)) - 2

and compare to zero:

x = 1 + 6 πn, x = 3 π + 1 + 6 πn

Periodicity of

2 csc (\frac{1}{3} (x - 1)) - 2 : 6 π

Combine period:

6 πn \leq x < 6 π + 6 πn

and undefined points.

The domain is

6 πn \leq x < 1 + 6 πn or 1 + 6 πn < x < 3 π + 1 + 6 πn or 3 π + 1 + 6 πn < x < 6 π + 6 πn

f (x) = \frac{3 x - 15}{x ^{2} - 25}

f (x) = x^{2} + 5 x

f (x) = \frac{1}{x} - 1

f (x) = 5 + 4 x - x^{2}

\frac{x + 2}{x ^{2} - 2 x - 3}

What is the domain of 2csc(1/3 (x-1))-2 ?
The domain of 2csc(1/3 (x-1))-2 is 6pin<= x<1+6pin\lor 1+6pin<x<3pi+1+6pin\lor 3pi+1+6pin<x<6pi+6pin