[Math] How to prove the domain of a function

Question

I'm given a function: $$f\left(x\right)=\left(\frac{1}{x-4}\right)$$
I have to find it's domain and prove it. Finding the domains is the easy part $(-\infty,4)\cup(4,\infty)$. But how do i prove it?

Answer 1

Let $D:=\mathbb{R}\backslash\{4\}$ and $x\in D$. Then $x-4\neq0$ so $(x-4)^{-1}$ exists in $\mathbb{R}$. Assuming $\operatorname{dom}(f)$ is to be as large as possible then we have $D\subseteq\operatorname{dom}(f)$. Also $4\not\in\operatorname{dom}(f)$ because in that case $x-4=0$ has no multiplicative inverse in $\mathbb{R}$. So $\operatorname{dom}(f)\subseteq D$. All in all, $\operatorname{dom}(f)=D$.