Answer
$\text{all real numbers}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
The domain of the given rational function, $
f(x)=\dfrac{2x+1}{x^2+3x+19}
,$ are the values of $
x
$ which will NOT make the denominator equal to $0.$
$\bf{\text{Solution Details:}}$
The denominator, $x^2+3x+19,$ is equivalent to $x(x+3)+19.$ By substituting any real value for $x,$ the expression $x(x+3)+19$ is always positive. Hence, the denominator is always a positive real number (i.e. the denominator never becomes $0.$)
Hence, the domain is the set of $
\text{all real numbers}
.$
