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If $f(x,y,z)=x^2+2xz+y$, determine $f([1,3]^3)$ and characterize this set in terms of openness, closedness, completeness, compactness and connectedness.

Since $[1,3]^3$ is compact then $f([1,3]^3)$ is, too. How would I go about expressing this function more precisely on this set?

It's obviously not open, closed, compact, complete because it's a closed subset of $\mathbb{R}^3$, and path connected meaning connected.

I think I can answer my own, just need confirmation from someone it's just $$[f(1),f(3)]$$ because of compactness and because it's connected.

Alex Ravsky
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Bozo Vulicevic
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    You might notice that it is monotonic in each variable separately. The maximum and minimum values are therefore trivial to compute. – copper.hat Jan 06 '15 at 00:10

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It seems the following.

The set $X=[0,3]^3$ is compact as a closed and bounded subset of $\Bbb R^n$ or because the segment $[0,3]$ is compact and its power is compact too by Tychonoff theorem. The set $X$ is linearly connected (and hence connected), because it is convex. The function $f$ is continuous as a composition of different continuous function. Therefore $f(X)$ is a compact connected subset of $\Bbb R$, that is a segment.

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Alex Ravsky
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