Question

Find the domain and range of f(x) = (1-2cosx)

Answer 1

Given, f(x) = 1/(1 - 2cosx) For defining the function , 1 - 2cosx ≠ 0we assume 1 - 2cosx = 0.Then, 1/2 = cosx ⇒x = π/3 and 5π/3 It means, in domain x ≠ π/3 and 5π/3 We consider interval ( 0, 2π)Then, domain ∈(0,π/3) ∪ (π/3 , 5π/3) ∪ (5π/3 , 2π)We know range is written as [least value, greatest value ]so, let's take value of f(x) in each interval .When x = 0 we have f(0) = 1/(1 -2cos(0)) = -1 it is maximum range of f(x) So, range always less than -1 y ≤ -1 Now, let's check f(π) = 1/(1 - 2cosπ) = 1/3 Because π Is in the interval (π/3, 5π/3), and it is minimum range of f(x) so, range of f(x) always greater than 1/3 Finally, range ∈(-∞ , -1] U [1/3, ∞)