# What is the domain of the function: f(x) = 5/(2x^2 - x - 3)?

Sep 15, 2015

${D}_{f} = \left(- \infty , - 1\right) \cup \left(- 1 , \frac{3}{2}\right) \cup \left(\frac{3}{2} , + \infty\right)$

#### Explanation:

We have to find the points of discontinuity:
$2 {x}^{2} - x - 3 = 0$
$2 {x}^{2} + 2 x - 3 x - 3 = 0$
$2 x \left(x + 1\right) - 3 \left(x + 1\right) = 0$
$\left(x + 1\right) \left(2 x - 3\right) = 0$
$x + 1 = 0 \vee 2 x - 3 = 0$
$x = - 1 \vee x = \frac{3}{2}$

${D}_{f} = \left(- \infty , - 1\right) \cup \left(- 1 , \frac{3}{2}\right) \cup \left(\frac{3}{2} , + \infty\right)$

Sep 15, 2015

${D}_{f} = \left(- \infty , - 1\right) \cup \left(- 1 , \frac{3}{2}\right) \cup \left(\frac{3}{2} , + \infty\right)$

#### Explanation:

First of All we need to know the Meaning of Domain:

A Domain is a set of inputs for which the function gives output
which is not indeterminate or invalid. For Example $\frac{1}{0}$. ,
$\frac{\infty}{\infty}$ etc are indeterminate or invalid forms .

You can find the list of indeterminate forms here .

In Other Words Domain is the set of values for which the function is defined.

Here the function ${F}_{x}$ is of the form $p / q$ a Rational function.

A Rational function is defined when
both $p$ and $q$ are defined and $q \notin 0$.

Considering the ${F}_{x}$ given.We have to find values where the ${F}_{x}$ gives valid output and which excludes the value where $q$ becomes $0$ and give rise to invalid form .

So
$q = 0$
when
$2 {x}^{2} - x - 3 = 0$
$2 {x}^{2} + 2 x - 3 x - 3 = 0$
$2 x \left(x + 1\right) - 3 \left(x + 1\right) = 0$
$\left(x + 1\right) \left(2 x - 3\right) = 0$
$x + 1 = 0 \vee 2 x - 3 = 0$
$x = - 1 \vee x = \frac{3}{2}$

${D}_{f} = \left(- \infty , - 1\right) \cup \left(- 1 , \frac{3}{2}\right) \cup \left(\frac{3}{2} , + \infty\right)$