1

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?

Amin Idelhaj
  • 1,732
G.Q
  • 37
  • 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, $\operatorname{dom}(f)=D$. – NeedForHelp Jan 26 '17 at 22:53
  • @NeedForHelp you should leave this as an answer – Stella Biderman Jan 26 '17 at 23:03
  • 3
    You can't have a function without already having the domain. What you have is an expression. Given an expression, we can then ask what is the largest subset of $\mathbb R$ or $\mathbb C$ for which the expression makes sense. This set is often called the "natural domain" associated with the expression. – zhw. Jan 26 '17 at 23:11
  • @zhw. relevant? https://math.stackexchange.com/questions/856226/how-do-you-prove-the-domain-of-a-function – BCLC Dec 01 '20 at 00:06

1 Answers1

4

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, $\operatorname{dom}(f)=D$.

NeedForHelp
  • 493