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Question: If the function f:[1 , ∞] → \[1, ∞) is defined by f(x) = 2<sup>x (x − 1)</sup> , then f<sup>−1</sup>...

If the function f:[1 , ∞] → [1, ∞) is defined by f(x) = 2x (x − 1) , then f−1 (x) is (1999 – 2Marks)

A

(1/2)x(x − 1)

B

−13≤y<1- \frac{1}{3} \leq y < 1

C

−13>y>−1- \frac{1}{3} > y > - 1

D

Not defined

Answer

−13≤y<1- \frac{1}{3} \leq y < 1

Explanation

Solution

y = 2x(x − 1) ⇒ x2 – x – log2y = 0;

Since x is + ive, we choose only + out of ±\pm

∴ x=12(1+1+4log⁡2y)x = \frac { 1 } { 2 } \left( 1 + \sqrt { 1 + 4 \log _ { 2 } y } \right) or f−1(x)=12(1+1+4log⁡2x)f ^ { - 1 } ( x ) = \frac { 1 } { 2 } \left( 1 + \sqrt { 1 + 4 \log _ { 2 } x } \right)