Restricting the domain of a function
The domain of a function, if not specified, is taken to be its
natural domain, but in many applications we may want to restrict it to a range of values corresponding to a real situation. For instance, if $C(x) = x^2+5$ represents the cost of produce $x$ kilograms of ectoplasm essence in a day, the function has no real meaning if $x$ is negative, so we should really restrict the domain to $[0,\infty)$, or, if the formula does not apply in the case when no ectoplasm essence is made, to $(0,\infty)$. Further, if it is not possible to make more than 20 kilograms in a day, then we should further restrict the domain to $[0, 20]$ or $(0,20]$.
We represent these possibilities graphically below:
#[Examples (based on above scenario)][Ejemplos (basados en el escenario anterior)]#
In the following graphs, a solid dot at an endpoint in a graph indicates that its $x$ value is in the domain, so the point is actually a point on the graph. An open dot indicates that its $x$ value is in not in the domain, so the point is actually deleted from the graph.
#[Domain][Dominio]#: $[0,\infty)$
#[$0$ in the domain][$0$ incido en el dominio]#
#[Domain][Dominio]#: $(0,\infty)$
#[$0$ not in the domain][$0$ no en el dominio]#
#[Domain][Dominio]#: $[0,20]$
#[$0$ in the domain][$0$ incido en el dominio]#>
#[Domain][Dominio]#: $(0,20]$
#[$0$ not in the domain][$0$ no incluido en el dominio]#