Question

Find the domain of the rational function.

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

Short answer

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

Step-by-step solution

Step 1. Given information

The given function is$\hspace{0.17em}F\left(x\right)=\frac{3x\left(x-1\right)}{2x^2-5x-3}$

Step 2. Find the domain    

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

So,

$$\begin{gathered} 2x^2-5x-3\ne 0 \\ 2x^2-6x+x-3\ne 0 \\ 2x(x-3)+(x-3)\ne 0\left[taking2xand1outascommon\right] \\ \left(2x+1\right)\left(x-3\right)\ne 0\left[takingx-3outasacommon\right] \\ 2x+1\ne 0 \\ 2x\ne \frac{-1}{2} \\ or \\ x\ne 3 \end{gathered}$$

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

Therefore, the domain islocalid="1646075792279" $\left(-\infty \hspace{0.17em},\hspace{0.17em}-\frac{1}{2}\right)\cup \left(-\frac{1}{2},\hspace{0.17em}3\right)\cup \left(3,\hspace{0.17em}\infty \hspace{0.17em}\right)$