How to correctly represent a union of two surfaces in $\mathbb{R}^{n}$?

Question

Suppose I have two smooth surfaces $\Gamma_{1}$ and $\Gamma_{2}$ written as graphs over variables $x \in U \subset \mathbb{R}^{n}$, and these surfaces are symmetric in some sense i.e.

$$\Gamma_{1} = \{(y, x) \in \mathbb{R}^{n+1}: y = f(x) \}$$

$$\Gamma_{2} = \{(y,x) \in \mathbb{R}^{n+1}: y = -f(x) \}$$

Is it correct to write their union $\Gamma_{1} \cup \Gamma_{2}$ in the following way:

$$\Gamma_{1} \cup \Gamma_{2} = \{(y, x) \in \mathbb{R}^{n+1}: y^{2} = f^{2}(x) \}$$

where the suberscript represents the square of the respective function?

Answer 1

$\newcommand{\Reals}{\mathbf{R}}$Yes, if $f$ is a real-valued function in some set $U$, then the equation $y^{2} = f(x)^{2}$ for $(y, x)$ in $\Reals \times U$ is equivalent to $$ 0 = y^{2} - f(x)^{2} = \bigl(y - f(x)\bigr)\bigl(y + f(x)\bigr),\quad (y, x) \in \Reals \times U, $$ i.e., to $y = f(x)$ or $y = -f(x)$.