Question

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

Answer 1

`D_f=(-oo,-1)uu(-1,3/2)uu(3/2,+oo)`

Explanation:

We have to find the points of discontinuity:
`2x^2-x-3=0`
`2x^2+2x-3x-3=0`
`2x(x+1)-3(x+1)=0`
`(x+1)(2x-3)=0`
`x+1=0 vv 2x-3=0`
`x=-1 vv x=3/2`

`D_f=(-oo,-1)uu(-1,3/2)uu(3/2,+oo)`

Answer 2

`D_f=(-oo,-1)uu(-1,3/2)uu(3/2,+oo)`

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 `1/0`. ,
`oo/oo` 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 !in0`.

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
`2x^2-x-3=0`
`2x^2+2x-3x-3=0`
`2x(x+1)-3(x+1)=0`
`(x+1)(2x-3)=0`
`x+1=0 vv 2x-3=0`
`x=-1 vv x=3/2`

`D_f=(-oo,-1)uu(-1,3/2)uu(3/2,+oo)`