# Chapter 7 - Rational Functions - 7.1 Rational Functions and Variation - 7.1 Exercises: 41

$\text{all real numbers}$

#### 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 $\text{all real numbers} .$

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