Let f: R → R be the function defined by f(x) = 1/(2 – cos x) ∀ x

1.	Let f: R → R be the function defined by f(x) = 1/(2 – cos x) ∀ x ∈ R. Then, find the range of f.
Answer» 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].

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

Answer»

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].