The range of the function f ( x )=√4- x 2+√ x 2-1 is

Question

The range of the function f ( x )=√4- x 2+√ x 2-1 is (A) [√3, √7] (B) [√3, √5] (C) [√2, √3] (D) [√3, √6]

Tardigrade Admin · Accepted Answer

Let x2=4 cos 2 θ+ sin 2 θ then (4-x2)=3 sin 2 θ and (x2-1)=3 cos 2 θ ∴ f ( x )=√3| sin θ|+√3| cos θ| ⇒ y min .=√3 text and y max =√3((1/√2)+(1/√2))=√6 . Hence range of f ( x ) is [√3, √6] Ans. ]

Q. The range of the function $f (x) = 4 - x^{2} + x^{2} - 1$ is

$[3, 7]$

$[3, 5]$

$[2, 3]$

$[3, 6]$

Let $x^{2} = 4 cos^{2} θ + sin^{2} θ$
then $(4 - x^{2}) = 3 sin^{2} θ$ and $(x^{2} - 1) = 3 cos^{2} θ$
$∴ f (x) = 3 ∣ sin θ ∣ + 3 ∣ cos θ ∣$
$\Rightarrow y_{m i n .} = 3 and y_{m a x} = 3 (\frac{1}{2} + \frac{1}{2}) = 6 .$
Hence range of $f (x)$ is $[3, 6]$ Ans. $]$

Q. The range of the function f(x)=4−x2​+x2−1​ is

[3​,7​]

[3​,5​]

[2​,3​]

[3​,6​]