Question

What is the domain and range of `1 / (x^2 + 5x + 6)`?

Answer 1

The domain is `x in (-oo,-3)uu(-3, -2)uu(-2, +oo)`. The range is `y in (-oo,-4]uu [0, +oo)`

Explanation:

The denominator is

`x^2+5x+6=(x+2)(x+3)`

As the denominator must be `!=0`

Therefore,

`x!=-2` and `x!=-3`

The domain is `x in (-oo,-3)uu(-3, -2)uu(-2, +oo)`

To find the range, proceed as follows :

Let `y=1/(x^2+5x+6)`

`y(x^2+5x+6)=1`

`yx^2+5yx+6y-1=0`

This is a quadratic equation in `x` and the solutions are real only if the discriminant is `>=0`

`Delta=b^2-4ac=(5y)^2-4(y)(6y-1) >=0`

`25y^2-24y^2+4y>=0`

`y^2+4y>=0`

`y(y+4)>=0`

The solutions of this inequality is obtained with a sign chart.

The range is `y in (-oo,-4]uu [0, +oo)`

graph{1/(x^2+5x+6) [-16.26, 12.21, -9.17, 5.07]}