The domain of \(f(x)=\cos ^{-1}\left(\frac{x-3}{2}\right)-\log _{10}(4-x)\) is
The domain of \(f(x)=\cos ^{-1}\left(\frac{x-3}{2}\right)-\log _{10}(4-x)\) is
\((1,4)\)
$[1,4)$
$[1,4)$
\([1,4]\)
Solution
Let \(f(x)=\cos ^{-1}\left(\frac{x-3}{2}\right)-\log _{10}(4-x)\)
For domain of \(\log _{10}(4-x) \Rightarrow 4-x > 0 \Rightarrow x < 4\)
For domain of \(\cos ^{-1}\left(\frac{x-3}{2}\right) \Rightarrow-1 \leq \frac{x-3}{2} \leq 1\)
\(\Rightarrow \quad 1 \leq x \leq 5\)
Now domain of $f(x)$ will be $x \in [1,4)$