If the function f:[1,∞)→[1,∞) is defined by f(x)=2x(x−1), then f−1(x) is
A
(12)x(x−1)
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B
12(1+√1+4log2x)
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C
12(1−√1+4log2x)
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D
not defined
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Solution
The correct option is B12(1+√1+4log2x) Let y=2x(x−1), where y≥1 as x≥1 Taking log2 on both sides, we get log2y=log22x(x−1) ⇒log2y=x(x−1) ⇒x2−x−log2y=0 ⇒x=1±√1+4log2y2 For y≥1,log2y≥0⇒4log2y≥0⇒1+4log2y≥1 ⇒√1+4log2y≥1 ⇒−√1+4log2y≤−1 ⇒1−√1+4log2y≤0 But x≥1 So, x=1−√1+4log2y is not possible. Therefore, we take x=12(1+√(1+4log2y)) ⇒f−1(y)=12(1+√1+4log2y) ⇒f−1(x)=12(1+√1+4log2y)