f : [2, ∞)→(−∞, 4

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Target Exam

CUET

Subject

-- Mathematics - Section B1

Chapter

Relations and Functions

Question:

f : [2, ∞)→(−∞, 4] , where f(x) = x(4 − x), then $f^{−1}(x)$ is given by

Options:

$2-\sqrt{4-x}$

$2+\sqrt{4-x}$

$-2+\sqrt{4-x}$

$-2-\sqrt{4-x}$

Correct Answer:

$2+\sqrt{4-x}$

Explanation:

$f (x) = x(4 − x) ⇒ f (f^{-1}(x))= 4f^{-1}(x)− (f^(x))^2 = x$

$(f^{-1}(x))^2-4f^{-1}(x)+x=0⇒f^{-1}(x)=\frac{4±\sqrt{16-4x}}{2}=2±\sqrt{4-x}$

But $f^{-1}: (∞, 4] [2, ∞) ⇒ f^{-1}(x) = 2+\sqrt{4-x}$