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  4. The domain of definition of the function f(x) = ( log(2x -3)/√x-1) + √5 - 2x is

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Q. The domain of definition of the function $f\left(x\right) = \frac{\log\left(2x -3\right)}{\sqrt{x-1}} + \sqrt{5 - 2x} $ is

COMEDKCOMEDK 2008Relations and Functions - Part 2

Solution:

$f\left(x\right) = \frac{\log\left(2x -3\right)}{\sqrt{x-1}} + \sqrt{5 - 2x} $
$ \sqrt{x-1} > 0 \Rightarrow x > 1$ and $ \log\left(2x - 3\right) > 0 $
$ \Rightarrow 2x - 3 > 0 \Rightarrow x > \frac{3}{2} $
and $ \sqrt{5 - 2x} \ge 0 \Rightarrow 5 \ge 2x \Rightarrow x \le \frac{5}{2} $
$ \therefore \, \, \frac{3}{2} < x \le \frac{5}{2} $
Hence, domain of $ f\left(x\right) $ is $\left( \frac{3}{2} , \frac{5}{2}\right]$