Question

Find the range of the following functions given by f(x) = 1 + 3 cos2x

(Hint: –1 ≤ cos 2x ≤ 1 ⇒ –3 ≤ 3 cos 2x ≤ 3 ⇒ –2 ≤ 1 + 3cos 2x ≤ 4)

Answer 1

We know the value of cos 2x lies between –1, 1

So –1 ≤ cos 2x ≤ 1

Multiplying by 3, we get

–3 ≤ 3cos 2x ≤ 3

Adding 1, we get

–2 ≤ 1 + 3cos 2x≤ 4

Or, –2 ≤ f(x) ≤ 4

Hence f(x) ∈ [–2, 4]

Therefore, the range of f = [–2, 4]