Question

The domain of $\frac{1}{\sqrt{x-x^2}}+\sqrt{3x-1-2x^2}$ is:

Answer 1

Consider the problem

$\begin{array}{c}x-x^2=0\\ \Rightarrow x\left(1-x\right)=0\\ \Rightarrow x=0,1\end{array}$

And

$\begin{array}{c}3x-1-2x^2=0\\ \Rightarrow -2x^2+3x-1=0\\ \Rightarrow 2x^2-3x+1=0\\ \Rightarrow 2x^2-2x-x+1=0\\ \Rightarrow 2x\left(x-1\right)-1\left(x-1\right)=0\\ \Rightarrow \left(x-1\right)\left(2x-1\right)=0\\ \Rightarrow x=1,x=\frac{1}{2}\end{array}$

$3x-1-2x^2<0whenx\in \left(-\infty ,2\right)\cup \left(1,\infty \right)$

and $x-x^2\le 0whenx\in \left(-\infty ,0\right]\cup \left[1,\infty \right)$

Therefore, Domain of given function is

$R-\left\{\left\{\left(-\infty ,2\right)\cup \left(1,\infty \right)\right\}\cup \left\{\left(-\infty ,0\right]\cup \left[1,\infty \right)\right\}\right\}$