  In this paper we study the following type of functions $f:
\mathcal{Q}_{\mathbb{R}_{3}} \to \mathbb{R}_{3}$, where $
\mathcal{Q}_{\mathbb{R}_3}$ is the quadratic cone of the algebra
$\mathbb{R}_{3}$. From the fact that it is possible to write the algebra $
\mathbb{R}_{3}$ as a direct sum of quaternions, we get the observation that it
is possible to find a clever representation for $ \mathcal{Q}_{\mathbb{R}_3}$.
By using this result a slice regular theory was introduced and a Cauchy formula
is discussed. Moreover, a detailed study of the zeros is performed. Finally, we
find a formula for the determinant of a matrix with entries in $
\mathcal{Q}_{\mathbb{R}_3}$.
