Question

Let f: R → R be the function defined by f(x) = `1/(2 - cosx)` ∀ x ∈ R.Then, find the range of f

Answer 1

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]`.