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  4. If the function f:[1, ∞) arrow[1, ∞) is defined by f(x)=2x(x-1) then f-1(x) is

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Q. If the function $f:[1, \infty) \rightarrow[1, \infty)$ is defined by $f(x)=2^{x(x-1)}$ then $f^{-1}(x)$ is

Relations and Functions - Part 2

Solution:

Let $f(x)=y=2^{x(x-1)}$
$\Rightarrow \log y=x(x-1) \log ^2$
$\Rightarrow x^2-x-\log _2 y=0$
$\Rightarrow x=\frac{1 \pm \sqrt{1+4 \log _2 y}}{2}$
$\Rightarrow x=f^{-1}(y)=\frac{1 \pm \sqrt{1+4 \log _2 y}}{2}$
$\Rightarrow f^{-1}(x)=\frac{1+\sqrt{1+4 \log _2 y}}{2} \text { as } f^{-1}(x) \geq 1 .$