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Question
Let
\[A = \left\{ x \in R : x \geq 1 \right\}\] The inverse of the function,
\[f : A \to A\] given by
\[f\left( x \right) = 2^{x \left( x - 1 \right)} , is\]
Options
\[\left( \frac{1}{2} \right)^{x \left( x - 1 \right)}\]
\[\frac{1}{2} \left\{ 1 + \sqrt{1 + 4 \log_2 x} \right\}\]
\[\frac{1}{2} \left\{ 1 - \sqrt{1 + 4 \log_2 x} \right\}\]
not defined
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Solution
\[\text{Let} f^{- 1} \left( x \right) = y . . . \left( 1 \right)\]
\[ \Rightarrow f\left( y \right) = x\]
\[ \Rightarrow 2^{y\left( y - 1 \right)} = x\]
\[ \Rightarrow 2^{y^2 - y} = x\]
\[ \Rightarrow y^2 - y = \log_2 x\]
\[ \Rightarrow y^2 - y + \frac{1}{4} = \log_2 x + \frac{1}{4}\]
\[ \Rightarrow \left( y - \frac{1}{2} \right)^2 = \frac{4 \log_2 x + 1}{4}\]
\[ \Rightarrow y - \frac{1}{2} = \pm \frac{\sqrt{4 \log_2 x + 1}}{2}\]
\[ \Rightarrow y = \frac{1}{2} \pm \frac{\sqrt{4 \log_2 x + 1}}{2}\]
\[ \Rightarrow y = \frac{1}{2} + \frac{\sqrt{4 \log_2 x + 1}}{2} \left( \because y \geq1 \right)\]
\[So, f^{- 1} \left( x \right) = \frac{1}{2}(1 + \sqrt{1 + 4 \log_2 x} ) [\text{from}\left( 1 \right)]\]
So, the answer is (b).
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