Restriction of two variable function on curve

Question

Definition of restriction

Let $f:\mbox{Dom}(f)\subseteq\mathbb{R}^{2}\to\mathbb{R}$ be a function, $A\subseteq\mbox{Dom}(f)$ be a non empty set. The restriction of $f$ on $A$ is the two variable function $f_{|A}:A\to\mathbb{R}$ such that

$$f_{|A}(x,y)=f(x,y) \ \ \ \forall (x,y)\in A$$

moreover $\mbox{Gr}(f_{|A})=\left\{(x,y,z)\in\mathbb{R}^{3}\ \mbox{t.c.}\ (x,y)\in A, z=f(x,y) \right\}$

Restriction of a function on a graph of one variable function

This definition is not from a book, it is my reworking of theory (just for fun), so it could be pointless.

Let:

$\bullet \ \ \ f:\mbox{Dom}(f)\subseteq\mathbb{R}^2\to\mathbb{R}$ be a two variable function;

$\bullet \ \ \ g:\mbox{Dom}(g)\subseteq\mathbb{R}\to\mathbb{R}$ be a one variable function;

$\bullet \ \ \ A=\{(x,y)\in\mbox{Dom}(f) \ \mbox{s.t.} \ y=g(x)\}$

The restriction of $f$ on $A$ is the function

$$f_{|A}(x,y)=f(x,g(x))=h(x)$$

with domain

$$\mbox{Dom}(h)=\left\{x\in\mathbb{R} \ \mbox{s.t.} \ x\in\mbox{Dom}(g)\wedge (x,g(x))\in\mbox{Dom}(f)\right\}$$

My questions

1. From the definition of restriction, $h(x)$ is formally a two variable function, isn't it? Moreover is that true that the graph of $h$ is

$$\mbox{Gr}(h)=\left\{(x,y,z)\in\mathbb{R}^{3} \ \mbox{t.c.} \ x\in\mbox{Dom}(h),\, y=g(x),\, z=h(x)\right\}?$$

2. Can I say that $h(x)$ is a one variable function? If so, I think I can define the graph of $h$ on $Oxz$ as follow: $$\mbox{Gr}(h)=\left\{(x,z)\in\mathbb{R}^{2} \ \mbox{t.c.} \ x\in\mbox{Dom}(h), z=h(x)\right\}$$ right?

I'm really confused.

Answer 1

Ok, I'll try to answer my own questions. Please, check that it is correct.

In strict terms, $f_{|A}(x,y)$ is function of two variables $(x,y)=(x,g(x))$, however $y$ is not an indipendent variable: its value depends on $x$.

In addition, the graph of $f_{|A}$ is a subset of $\mathbb{R}^{3}$.

The function $h:\mbox{Dom}(h)\to\mathbb{R}$ is not the restriction of $f$ to $A$, but the function of one variable which associates $x\in\mbox{Dom}(h)$ the real number $h(x)=f(x,g(x))$, and it's not equivalent to $f_{|A}(x,y)$: their graphs aren't even comparable.