Question:
Let f: R → R be the function defined by f(x) = 1/(2 – cos x) ∀ x ∈ R. Then, find the range of f.
Solution:
Given,
f(x) = 1/(2 – cos x) ∀ x ∈ R
Let y = 1/(2 – cos x)
2y – ycos x = 1
cos x = (2y – 1)/ y
cos x = 2 – 1/y
Now, we know that -1 ≤ cos x ≤ 1
So,
-1 ≤ 2 – 1/y ≤ 1
-3 ≤ – 1/y ≤ -1
1 ≤ – 1/y ≤ 3
1/3 ≤ y ≤ 1
Thus, the range of the given function is [1/3, 1].