Let f : [1/2, 1] → [-1, 1] is

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Target Exam

CUET

Subject

-- Mathematics - Section B1

Chapter

Relations and Functions

Question:

Let f : [1/2, 1] → [-1, 1] is given by $f^{-1}(x) = 4x^3− 3x$

Options:

$\cos \left(\frac{1}{3} \cos ^{-1} x\right)$

$3 \cos \left(\cos ^{-1} x\right)$

$3 \sin ^{-1}(\sin x)$

$\sin \left(\frac{1}{3} \sin ^{-1} x\right)$

Correct Answer:

$\cos \left(\frac{1}{3} \cos ^{-1} x\right)$

Explanation:

$y=4x^3-3x$ let $x=\cos θ$

$y=4\cos^3 θ-3\cos θ$

$⇒θ\cos^{-1}(x)$

$y=\cos 3θ⇒\frac{1}{3}\cos^{-1}y=θ$

so $\cos\left(\frac{1}{3}\cos^{-1}y\right)=x$

$f^{-1}x=\cos \left(\frac{1}{3} \cos ^{-1} x\right)$