Question

Find the domain of the function $f\left(x\right)={\cos }^{-1}\left(\frac{6-3x}{4}\right)+cose{c}^{-1}\left(\frac{x-1}{2}\right)$

Answer 1

$\begin{array}{cc}f\left(x\right)={\cos }^{-1}\left(\frac{6-3x}{4}\right)+\cos e{c}^{-1}\left(\frac{x-1}{2}\right)& \\ \Rightarrow -1\le \frac{6-3x}{4}\le 1& \Rightarrow \frac{x-1}{2}\ne 1,x\ne 3\\ \Rightarrow -4\le 6-3x\le 4& \Rightarrow or\frac{x-1}{2}\ne -1,x\ne -1\\ \Rightarrow -10\le -3x\le -2& \\ \Rightarrow 10\ge 3x\ge 2& \\ \Rightarrow \frac{10}{3}\ge x\ge \frac{2}{3}& \end{array}$

Domain of function

$\left[\frac{2}{3]},\frac{10}{3}\right]-\left[3\right]$