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Question
Let f: R → R be the function defined by f(x) = `1/(2 - cosx)` ∀ 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]`.
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