set theoretical image of $\operatorname {Spec} k[x_1,\dots x_n]_{(x_1,\dots x_n)}$ We have the set theoretical image of $\operatorname{Spec} k[x_1,\dots x_n]_{(x_1,\dots x_n)}\to\operatorname{Spec} k[x_1,\dots x_n]$ defined by the canonical inclusion.
Is it constructible for a field $k$?
(Eisenbud & Harris, The Geometry of schemes, p210)
 A: Your question has been answered by user15654, but you may want to think about this question some more, and try to develop some geometric intuition for it.
Start with  the case $n = 1$.  What are the constructible subsets of Spec $k[x]$?  What is the image of Spec $k[x]_{(x)}$?
Now consider the case $n = 2$.  What are the points in Spec $k[x_1,x_2]$?  Which
sets of them are constructible?  What is the image of Spec $k[x_1,x_2]_{(x_1,x_2)}?  
There are very nice visualizations for these questions, which you can develop with a little thought and practice. 
A: The set-theoretic image of $\text{Spec }k[x_1,...,x_n]_{(x_1,...,x_n)}$ inside $\text{Spec }k[x_1,...,x_n] = \mathbb{A}^n_k$ is not constructible. This follows from the following facts:
1) Every constructible (or even locally constructible) subset of a Jacobson scheme is Jacobson
2) $\mathbb{A}^n_k$ is Jacobson
3) $\text{Spec }k[x_1,...,x_n]_{(x_1,...,x_n)}$ is not Jacobson
To see why (1) holds, note that an equivalent characterization of being Jacobson is that every locally closed set contains a closed point.
