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  4. Number of integers in the range of function f(x)= log (1/2)(x-(1/2))+ log 2(√4 x2-4 x+1) is (are)

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Q. Number of integers in the range of function $f(x)=\log _{\frac{1}{2}}\left(x-\frac{1}{2}\right)+\log _2\left(\sqrt{4 x^2-4 x+1}\right)$ is (are)

Relations and Functions - Part 2

Solution:

$ f(x)=\log _2\left(x-\frac{1}{2}\right)^{-1}+\log _2\left(\sqrt{(2 x-1)^2}\right) $
$f(x)=-\log _2\left(x-\frac{1}{2}\right)+\log _2(2 x-1) $
$f(x)=-\log _2\left(x-\frac{1}{2}\right)+\log _2\left(2\left(x-\frac{1}{2}\right)\right) $
$f(x)=-\log _2\left(x-\frac{1}{2}\right)+\log _2 2+\log _2\left(x-\frac{1}{2}\right) $
$f(x)=1 \text {, so the range of } f(x) \text { is }\{1\} $