Question

Find the domain of the rational function.

$Q\left(x\right)=\frac{-x\left(1-x\right)}{3x^2+5x-2}$

Short answer

The domain of the given function is

$\left(-\infty \hspace{0.17em},\hspace{0.17em}-2\right)\cup \left(-2,\hspace{0.17em}\frac{1}{3}\right)\cup \left(\frac{1}{3},\hspace{0.17em}\infty \hspace{0.17em}\right)$

Step-by-step solution

Step 1. Given information 

The given function is$Q\left(x\right)=\frac{-x\left(1-x\right)}{3x^2+5x-2}$

Step 2. Find the domain 

For the domain, the denominator should not be equal to zero.

So,

$$\begin{gathered} 3x^2+5x-2\ne 0 \\ 3x^2+6x-x-2\ne 0 \\ 3x(x+2)-(x+2)\ne 0\left[takingout3xand-1ascommon\right] \\ \left(3x-1\right)\left(x+2\right)\ne 0\left[takingoutx+2ascommon\right] \\ 3x-1\ne 0 \\ 3x\ne 1 \\ x\ne \frac{1}{3} \\ or \\ x+2\ne 0 \\ x\ne -2 \end{gathered}$$

Thus, the domain of the given function is all real numbers except $-2$and $\frac{1}{3}$.

Therefore, the domain is$\left(-\infty \hspace{0.17em},\hspace{0.17em}-2\right)\cup \left(-2,\hspace{0.17em}\frac{1}{3}\right)\cup \left(\frac{1}{3},\hspace{0.17em}\infty \hspace{0.17em}\right)$