What is the q(x)=csc(x)(sin(x)+tan(x))-1 ?

The q(x)=csc(x)(sin(x)+tan(x))-1 is

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q(x)=csc(x)(sin(x)+tan(x))-1

q (x) = csc (x) (sin (x) + tan (x)) - 1

Period : 2 π

Domain : 2 πn < x < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < x < π + 2 πn or π + 2 πn < x < \frac{3 π}{2} + 2 πn or \frac{3 π}{2} + 2 πn < x < 2 π + 2 πn

Range : f (x) < - 1 or f (x) > 1

Intercepts : None

Asymptotes : Vertical x = \frac{3 π}{2} + 2 πn, x = \frac{π}{2} + 2 πn

Extreme Points : None

Interval Notation

Domain : (2 πn, \frac{π}{2} + 2 πn) \cup (\frac{π}{2} + 2 πn, π + 2 πn) \cup (π + 2 πn, \frac{3 π}{2} + 2 πn) \cup (\frac{3 π}{2} + 2 πn, 2 π + 2 πn)

Range : (- \infty, - 1) \cup (1, \infty)

Solution steps

Periodicity of

csc (x) (sin (x) + tan (x)) - 1 : 2 π

Domain of

csc (x) (sin (x) + tan (x)) - 1 : 2 πn < x < \frac{π}{2} + 2 πn or \frac{π}{2} + 2 πn < x < π + 2 πn or π + 2 πn < x < \frac{3 π}{2} + 2 πn or \frac{3 π}{2} + 2 πn < x < 2 π + 2 πn

Range of

csc (x) (sin (x) + tan (x)) - 1 : f (x) < - 1 or f (x) > 1

Axis interception points of

csc (x) (sin (x) + tan (x)) - 1 :

None

Asymptotes of

csc (x) (sin (x) + tan (x)) - 1 :

Vertical

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

Extreme Points of

csc (x) (sin (x) + tan (x)) - 1 :

None

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f (x) = 4 - tan (x^{2} + 1)

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

f (x) = arccos (e^{3 x})