# Math Grade 11 Tangents 1) State domain, range, period, vertical asymptotes, zeros, symmetry and y-intercept of: y = -2tan(3x + 180°) + 3. 2) Sketch the graph y = 1/(tan(x)), and state its properties. if you cannot sketch the graph and show me, then please just state its properties, such as domain, range, period, vertical asymptotes, zeros, symmetry and y-intercept i would deeply appreciate if you could answer both questions. thank you.

You need to evaluate the domain of the given function using the condition that the argument of tangent function belongs to the interval `(-pi/2,pi/2)`  such that:

`-pi/2 < 3x + pi < pi/2 > -pi/2 - pi < 3x < pi/2 - pi-3pi/2 < 3x < -pi/2`

Multiplying by -1 yields:

`3pi/2...

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You need to evaluate the domain of the given function using the condition that the argument of tangent function belongs to the interval `(-pi/2,pi/2)`  such that:

`-pi/2 < 3x + pi < pi/2 > -pi/2 - pi < 3x < pi/2 - pi-3pi/2 < 3x < -pi/2`

Multiplying by -1 yields:

`3pi/2 > 3x > pi/2 => pi/2 > x > pi/6`

Hence, evaluating the domain of the given function yields `x in (pi/6 , pi/2).`

You need to remember that the range of tangent function is R.

You need to notice that the function has vertical asymptotes at `x = pi/6`  and `x = pi/2`  such that:

`lim_(x->pi/6, x>pi/6) (-2tan(3x + 180^o) + 3) = oo`

`lim_(x->pi/2, x>pi/2) (-2tan(3x + 180^o) + 3) = -oo`

You need to remember that the graph of the function intercepts x axis at y = 0, such that:

`-2tan(3x + 180^o) + 3 = 0 => -2tan(3x + 180^o)= -3`

`tan(3x + 180^o) = 3/2 => 3x + 180^o = arctan(3/2) + n*pi`

`3x = arctan(3/2) + n*pi - pi (180^o = pi)`

`x= arctan(3/2)/3 + n*pi/3 - pi/3`

Hence, evaluating the general solution to the equation of the given function yields `x = arctan(3/2)/3 + n*pi/3 - pi/3` .

You need to remember that the graph of the function intercepts y axis at x = 0, such that:

`-2tan(3*0 + 180^o) + 3 =y => y = -2tan 18o^o + 3 = y => y = 3`

Hence, evaluating the y intercept of the graph of the given function yields `(0,3).`

You need to find the period of the given tangent function, hence, you need to evaluate the ratio `pi/k` , where k represents the coefficient of the argument x, such that:

`p = pi/3 `

Hence, evaluating the period of the given function yields `pi/3.`

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