Question

The domain of the function $ f(x)=\sqrt{3-x}+{{\cos }^{-1}}\left( \frac{3-2x}{5} \right) $ , is

• A. $ [-1,\,\,3] $
• B. $ (-1,\,\,3] $
• C. $ [-1,\,\,3) $
• D. None of these

Answer 1

Answer: (A)

We have, the function $ f(x)=\sqrt{3-x}+{{\cos }^{-1}}\left( \frac{3-2x}{5} \right) $
For it's existence, $ 3-x\ge 0\Rightarrow x\le 3 $ ... (i)
and $ -1\le \frac{3-2x}{5}\le 1 $
$ \Rightarrow $ $ -5\le 3-2x\le 5 $
$ \Rightarrow $ $ -8\le -2x\le 2 $
$ \Rightarrow $ $ -1\le x\le 4 $ ... (ii)
From Eqs. (i) and (ii), it is clear that the domain of the given function $ f $ is
$ (-\infty ,\,\,3]\cap [-1,\,\,-4]=[-1,\,\,3] $