find the domain range of f, where..... (Calculus)

Question

Find the domain range of f, where.... (i). f(x) = (1)/(2-cos3x)..... (ii). f(x) = 0 (when x is rational); 1 (when x is irrational)....

Answer 1

(i)

The domain of f(x) is all values of x, since 2-cos3x can never be zero because cos is between -1 and 1. The range has to include 1/3 when cos3x=-1, 1/2 when cos3x=0, 1 when cos3x=1, so 1/3<f(x)<1. f(x) is continuous.

(ii)

f(x) is two lines which are not continuous because each line consists of a series of points. The domain is all real x. The function is not continuous, but of course has a range of 0 and 1 but nothing in between.