# Help please easy question!!?

Analyze the function f(x) = - 2 cot 3x. Include:
- Domain and range
- Period
asked Dec 27, 2012

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## Plots:

Source : http://www.wolframalpha.com

answered Jan 18, 2013

Let the function is y = f (x ) = -2 cot (3x ).

Compare the equation y = -2 cot (3x ) with y = a cot(bx - c ) where b > 0.

a = -2, b = 3 and c = 0.

First draw the graph of y = -2 cot (3x ).

The two consecutive vertical asymptotes of the graph y = a cot(bx - c ) can be found by solving the equations bx - c = 0 and bx - c = π.

∴ 3x = 0 and 3x = π.

x = 0 and x = π/3.

Therefore the two  consecutive  vertical  asymptotes  occur  at x = 0 and x = π/3.

The interval [0, π/3] corresponds to one cycle of the graph. The cycle begins with 0 and ends with π/3 and find the three middle values.

Between these two asymptotes x = 0 and x = π/3, plot a few points, including the x - intercept, as shown in the table.

 x y = -2 cot (3x) (x , y ) 0 y = -2 cot (3*0) = -2 cot0 = undefined (0 , ∞) π/6 y = -2 cot (3* π/6) = -2 cot(π/2) = -2 (0) = 0 (π/6 , 0) π/4 y = -2 cot (3* π/4) = -2 cot(135) = -2 (-1) = 2 (π/4 , 2) π/10 y = -2 cot (3* π/10) = -2 cot(54) = -2(0.72654) = -1.3084 (π/10, -1.3084) π/3 y = -2 cot (3* π/3) = -2 cot(π) = undefined (π/3, -∞)

First plotting the asymptotes.

The midpoint between two consecutive vertical asymptotes is an x - intercept of the graph. The period of the function y = -2 cot (3x) is the distance between two consecutive vertical asymptotes.

After plotting the asymptotes and the x - intercept, plot a few additional points between the two asymptotes and sketch one cycle. Finally, sketch one or two additional cycles to the left and right.

The domain of cotangent function, y = -2cot (3x) is - ∞ < x < ∞, where x not equal to integer multiplies of π/3 or x ≠ nπ/3 and x ≠ 0.

Observe the graph, the domain is R - {0, ±n(π/3)| n ∈ N} and range is the set of all real numbers R or (-∞, ).

Period is π / 3.

answered Jun 16, 2014