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$f(x) = 1 / (x-1)$ is not a function because for $x = 1$ there is a vertical asymptote which means infinte number of values of $y$ for $x = 1$.

It is a function for $\mathbf{R}- \{1\}$.

I want someone to just tell me if I am getting it right.

Walker
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ryan1
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    Yes... but I'll not say that there are an "infinite number of values for $x=1$". There are none. – Mauro ALLEGRANZA Sep 29 '20 at 14:59
  • Perhaps I'm being overly picky but I would say this sentence " for x=1 there is a vertical asymptote which means infinte number of values of y for x=1" is dead wrong. There are not an infinite number of values. There no values. Zero. – fleablood Apr 10 '21 at 21:31
  • A function needs a specified or implied domain. As $f(1)$ is undefined this is not a function $f: \mathbb R \to \mathbb R$. But it is a function $f:\mathbb R\setminus{1}\to \mathbb R$. – fleablood Apr 10 '21 at 21:33
  • Also note that $\bar f : \mathbb R \to \mathbb R$ defined by $\bar f(1)=1$ and $\bar f(x)=1/(x-1)$ for $x \neq 1$ is a function despite it having a vertical asymptote at $x=1.$ – md2perpe Apr 11 '21 at 00:07

2 Answers2

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When you define a function you need to specify what domain and codomain it has. The function cannot have this functional equation if the expression is not defined for a member of its domain (i.e. division by $0$). So you did not specify a function, you specified an equation that the function f is supposed to satisfy. There are multiple solutions to this equation so this does not define a single function. If you want to be more specific you could define $f$ to be the function with the maximal domain in $\mathbb R$ with codomain $\mathbb R$ that satisfies this equation. In this case, yes f is the function with domain $\mathbb R \setminus \{1\}$, codomain $\mathbb R$ and functional equation $f(x)=\frac{1}{x-1}$ for all $x$ of the domain.

L. t.
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It is a function, with an expected domain of $(-\infty, 1) \cup (1, +\infty)$ instead of all $\Bbb{R}$. When you define a function, you need to specify what your domain is.

user577215664
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aram10
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