# Is there a name for a position on a function surface and the value at that position?

Is there a name for a position on a function surface and the value at that position? E.g., if I have the function $f(x,y) = x^2 + y^2$, and I know that at the point $(2, 2)$ it evaluates to $8$, is there a name for the data structure $(2, 2, 8)$ [which is in this example $(x, y, f(x, y))$]?

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This doesn't quite answer your question, but the set of all points of the form $(x,y,f(x,y))$ is called the graph of the function $f$, so in your case, $(2,2,8)$ is a point in the graph. In general, for any function $f : X \to Y$ between two sets, the set $\{ (x,f(x)) \mid x \in X \} \subseteq X \times Y$ is the graph of $f$. –  fuglede Oct 5 '11 at 20:36
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## 1 Answer

It is an element of the graph of $f$. The formal definition of the graph of a function $f:X\to Y$, where $X$ and $Y$ are any two sets, is the subset $\Gamma(f)\subset X\times Y$ consisting of $$\Gamma(f)=\{(x,y)\in X\times Y\mid y=f(x)\}.$$ So, in your case, $X=\mathbb{R}^2$ is the plane, $Y=\mathbb{R}$ is the real numbers, and $$\Gamma(f)=\{((x,y),z)\in \mathbb{R}^2\times \mathbb{R}\mid z=f(x,y)\}.$$ But we usually identify $\mathbb{R}^2\times\mathbb{R}$ with $\mathbb{R}^3$, of course, and we get $$\Gamma(f)=\{(x,y,z)\in \mathbb{R}^3\mid z=f(x,y)\},$$ or in other words, $\Gamma(f)$ consists of the triples $(x,y,f(x,y))$, such as $(2,2,8)$.

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Thanks. I am guessing there is no fancy name for "element of the graph of f"? –  shino Oct 5 '11 at 21:24
No, I don't believe so. –  Zev Chonoles Oct 6 '11 at 2:06
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