State the domain and range for the function

* Domain: all reals except multiples of 2π

* Range: (-∞ , -2] ∪ [2 , ∞)

Step-by-step explanation:

* Lets revise the period, the domain and the range of csc(x)

- The period of csc(x) is 2π

- To find the period of csc(x) use 2π / coefficient of x

- The domain of csc(x) is all x ≠ nπ

- The range is y ≤ -1 , y ≥ 1

* Lets revise the vertical and the horizontal stretch and compress

- A vertical stretching is the stretching of the graph away from

the x-axis

• if k > 1, the graph of y = k•f(x) is the graph of f(x) vertically

 stretched by multiplying each of its y-coordinates by k.

- A vertical compression is the squeezing of the graph toward

 the x-axis.

• if 0 < k < 1 (a fraction), the graph is f (x) vertically compressed

  by multiplying each of its y-coordinates by k.

- A horizontal stretching is the stretching of the graph away from

 the y-axis

• if 0 < k < 1 (a fraction), the graph is f (x) horizontally stretched by

 dividing each of its x-coordinates by k.

- A horizontal compression is the squeezing of the graph toward

 the y-axis.

• if k > 1, the graph of y = f (k•x) is the graph of f (x) horizontally

 compressed by dividing each of its x-coordinates by k

∵ f(x) = 2csc(x/2)

- The coefficient of x is 1/2

∵ The period of x = 2π

∴ The period of x/2 = 2π/1/2 = 4π

∵ The domain of csc(x) is all x ≠ nπ

∴ The domain of csc(x/2) is all x ≠ n2π

∵ f(x) = 2csc(x/2)

∵ csc(x/2) multiplying by 2

- That means every y-coefficient multiplying by 2

∵ The range of csc(x/2) is y ≤ -1 and y ≥ 1

∴ The rang of f(x) = 2csc(x/2) is y ≤ -1(2) and y ≥ 1(2)

∴ The rang of f(x) = 2csc(x/2) is y ≤ -2 and y ≥ 2

* Domain: all reals except multiples of 2π

* Range: (-∞ , -2] ∪ [2 , ∞)

* Look to the graph attached

- The red is y = csc(x)

- The blue is f(x) = 2csc(x/2)

the line of