Help please easy question!!?

Question

Analyze the function f(x) = - 2 cot 3x. Include:
- Domain and range
- Period

Answer 1

Domain:

---

Range:

---

---

»

Number lines:

Domain:

---

Range:

 

Plots:

---

 

 

Source : http://www.wolframalpha.com
 

 

Answer 2

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.