You need to find the domain and the range of the given function `f(x) = 2 tan(9x + 180) + 9` , hence, you need to remember that the domain of the tangent function is the interval `(-pi/2,pi/2)` and the range is R.

Hence, since the argument of tangent function needs...

## View

This Answer NowStart your **48-hour free trial** to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

Already a member? Log in here.

You need to find the domain and the range of the given function `f(x) = 2 tan(9x + 180) + 9` , hence, you need to remember that the domain of the tangent function is the interval `(-pi/2,pi/2)` and the range is R.

Hence, since the argument of tangent function needs to be in interval `(-pi/2,pi/2)` yields:

`-pi/2 < 9x + 180 < pi/2`

You need to substitute `pi` for 180 such that:

`-pi/2 < 9x +pi < pi/2`

You need to subtract `pi` both sides such that:

`-pi - pi/2 < 9x < pi/2 - pi => -3pi/2 < 9x < -pi/2`

You need to divide by -9 such that:

`3pi/18 > x > pi/18 => x in (pi/18 , pi/6)`

Hence, the domain of the function is the interval `(pi/18 , pi/6)` and the range is the interval `(f(pi/18),f(pi/6)).`

You may find the period of tangent function checking the coefficient of x, hence, since the coefficient of x is 9, the period of tangent function is `pi/9` .

You need to find the zeroes of the given function, hence, you need to solve the equation `2 tan(9x + 180) + 9 = 0` such that:

`2 tan(9x + 180) + 9 = 0 => 2 tan(9x + 180)= -9`

`tan(9x + 180) = -9/2 => 9x + 180 = arctan(-9/2) + n*pi`

`9x = -arctan(9/2) + n*pi - 180 => 9x = -arctan(9/2) + n*pi - pi`

`x =-arctan(9/2) + n*pi/9 - pi/9`

You may find y intercept of the graph considering `x = 0` such that:

`f(0) = 2 tan(0 + 180) + 9 => f(0) = 2 tan 180 + 9 => f(0) = 2 tan pi + 9 =>f(0) = 2*0 + 9 =>f(0) = 9`

**Hence, evaluating the domain of the function yields interval `(pi/18 , pi/6)` and the range yields the interval `(f(pi/18),f(pi/6)), ` then, evaluating the period of tangent function yields `pi/9,` the zeroes of the given function yields `x = -arctan(9/2) + n*pi/9 - pi/9` and y intercept yields `f(0) = 9` .**