Question

Please help What is the equation for a cosecant function with vertical asymptotes found at D%5Cfrac%7B%5Cpi%20%7D%7B2%7D%20%2B%5Cfrac%7B%5Cpi%20%7D%7B2%7Dn" id="TexFormula1" title="x=\frac{\pi }{2} +\frac{\pi }{2}n" alt="x=\frac{\pi }{2} +\frac{\pi }{2}n" align="absmiddle" class="latex-formula"> such that n is an integer? f (x) = 2cscx g(x) = 4csc2x h(x) = 4csc3x $J(x)=2csc\frac{x}{2}$

Answer 1

The graph of csc(x) = 1/sin(x) has asymptotes at the points where sin(x) = 0, which happens for x = 2nπ and x = (2n + 1)π, where n is any integer.

This means

• f(x) = 2 csc(x) has the same asymptotes, at x = 2nπ and x = π + 2nπ

• g(x) = 4 csc(2x) has asymptotes at 2x = 2nπ and 2x = π + 2nπ, or equivalently x = nπ and x = π/2 + nπ

• h(x) = 4 csc(3x) has asymptotes at 3x = 2nπ and 3x = π + 2nπ, or x = 2nπ/3 and x = π/3 + 2nπ/3

• j(x) = 2 csc(x/2) has asymptotes at x/2 = 2nπ and x/2 = n + 2nπ, or x = 4nπ and x = 2π + 4nπ

so none of these choices are correct.