FunctionHard
Question
The range of the function f(x) = log√2 (2 - log2(16 sin2x + 1)) is
Options
A.(- ∞, 1)
B.(- ∞, 2)
C.(- ∞, 1]
D.(- ∞, 2]
Solution
Here (2 - log2 (16 sin2x + 1) > 0
⇒ 0 < 16 sin2x + 1 < 4
⇒ 0 ≤ sin2x < 3/16
⇒ 1 ≤ 16 sin2x + 1 ≤ 4
⇒ 0 ≤ log2 (16 sin2x + 1) < 2
⇒ 2 ≥ 2 - log2 (16 sin2x + 1) > 0
⇒ log√2 2 ≥ log√2 (2 - log2 (16 sin2x + 1)) > - ∞
⇒ 2 ≥ y > - ∞
Hence range is y ∈ (- ∞, 2]
⇒ 0 < 16 sin2x + 1 < 4
⇒ 0 ≤ sin2x < 3/16
⇒ 1 ≤ 16 sin2x + 1 ≤ 4
⇒ 0 ≤ log2 (16 sin2x + 1) < 2
⇒ 2 ≥ 2 - log2 (16 sin2x + 1) > 0
⇒ log√2 2 ≥ log√2 (2 - log2 (16 sin2x + 1)) > - ∞
⇒ 2 ≥ y > - ∞
Hence range is y ∈ (- ∞, 2]
Create a free account to view solution
View Solution FreeMore Function Questions
The function f(x) = cot-1 + cos-1 is defined on the set S, where S is equal to:...The function f : [−1/2, 1/2] → [−π/2, π/2] defined by f(x) = sin-1(3x − 4x3) is -...The minimum value of f(x) = | 3 - x | + | 2 + x | + | 5 - x | is -...The sum of all the elements in the range of $f(x) = Sgn(sinx) + Sgn(cosx) + Sgn(tanx) + Sgn(cotx)$, $x \neq \frac{n\pi}{...If f : R → R , f(x) = 2x ; g : R → R, g(x) = x + 1, then (f .g) (2) equals -...