Answer:
* Domain: all reals except multiples of 2π
* Range: (-∞ , -2] ∪ [2 , ∞)
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)