How to write this domain notation?

Question

For $y=1/x^3$ the domain is $\{x\mid x\neq 3\}$. In notation, would we write the domain as: $(3, +\infty]$ or $[4, \infty)$? Perhaps my inclusive brackets are not right but more so focused on if we would put 3 exclusive or 4?

CORRECTION: I meant to write y=1/(x-3) and yes I realize my domain notation was backwards/incorrect.

Answer 1

Note that $$\{x\in\mathbb{R}\mid x\neq 3\}=(-\infty,3)\cup(3,\infty).$$

Answer 2

We have: $$(\forall x \in \mathbb{R})(x \neq 3 \;\leftrightarrow\; x < 3 \;\mathrm{or} \;x > 3).$$

Hence: $$\{x \in \mathbb{R} : x \neq 3\} = \{x \in \mathbb{R}:x < 3\;\mathrm{or} \;x > 3\}.$$

But the right-hand side above can be rewritten like so:

$$\mathrm{RHS} = \{x \in \mathbb{R}:x<3\} \cup \{x \in \mathbb{R}:x>3\} = (-\infty,3) \cup (3,\infty).$$

Hence:

$$\{x \in \mathbb{R} : x \neq 3\} = (-\infty,3) \cup (3,\infty)$$

Answer 3

For $y=1/x^3$ the domain is $\{x\mid x\neq 3\}$. In notation, would we write the domain as: $(3, +\infty]$ or $[4, \infty)$? Perhaps my inclusive brackets are not right but more so focused on if we would put 3 exclusive or 4?

$(3; +\infty) = \{x\in\Bbb R\mid x< 3\}$

$[4; \infty)= \{x\in\Bbb R\mid x\leq 4\}$

Neither is correct, though the first is somewhat closer. (Note $3.5\neq 3$ and such.)

You wish to express $\{x\mid x<3~\lor~3<x\}$, which is the union $(-\infty;3)\cup(3;+\infty)$.

Also written as $\Bbb R\setminus\{3\}$.

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PS: For $y=1/x^3$ the domain is $\{x\in \Bbb R\mid x\neq 0\}$.   Are you sure you did not mean $y=1/(x-3)$?