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Given $ax+b$, we often draw a line $y = ax+b$ and we say this is for all values of $x$ and $y$, but is it wrong to say that $x$ is every value at the same time? For instance we say $x\in \mathbb{R}$. When we then say that $x=3$, are we introducing a second equation imposing limits on $x$? So $x$ must fulfill both $y = ax+b$ and $x=3$, meaning that $x \in \mathbb{R} \;\bigwedge\; x \in \{3\}$.

So is $x$ a value, or all values?

Frank Vel
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3 Answers3

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Too long for a comment, and there's already a good short answer.

This interesting question is an instance of a common misunderstanding that most students eventually resolve intuitively. An "$x$" sometimes means a particular number, usually one you must find, sometimes a typical number (or element of the domain of a function). The difference is rarely mentioned explicitly.

The $x$ in $y = mx +b$ is the second kind. What's really being specified is the function $f$ defined by the rule
$$ f(\text{anything}) = m \times \text{ anything} + b $$ - no need to mention $x$ or $y$. This is often written as

the function $f(x) = mx+b$

even though $f(x)$ isn't the function, $f$ is.

Should we ban that abuse of the language? That's a hard question to answer. Most of the time students can understand from the context what's going on. In those cases the extra cumbersome prose would be more confusing than helpful. But some of the time the abuse leads to confusion.

Ethan Bolker
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$x$ is a single value just like $a$ and $b$, but the line you plot shows you every value of $ax+b$ given every value of $x$.

The fact that $x$ is a variable with $x \in \mathbb R$ could be any real number, but to get the value of $y = ax + b$ you need to specify a fixed value of $x$, which gives you a fixed value of $y$.

naslundx
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I like to think of the variable $x$ in your equation as a set, and it all starts to make sense. What the function notation is really providing is a mapping from every member of the set $x$ to a member of the set $f(x)$. We can specify an $x$ value because we are merely choosing a value from the set $x$ and finding the mapped member in $f(x)$

Brevan Ellefsen
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  • If thinking of $x$ here as a set helps you, then do that. But in general it may cause more confusion than it avoids, since it doesn't give you a way to think of the value of $f$ at a typical but otherwise unspecified point. And I think the comment on the axiom of choice here is unnecessary (almost all the time in practice) and potentially confusing. – Ethan Bolker Jul 06 '16 at 15:28
  • @EthanBolker I see it as a matter of taste. I can more easily think of the value of $f$ at a typical point (not sure what you even mean by that... the average value of $f$? The most common value of $f$? Any given value of $f$ in the range?) when expressed as a set, because it helps me to think of continuous functions as a set of discrete points... for example, the $x$ axis doesn't make much sense to me until I see it as a representation of the set $\mathbb{R}$ where each point on the line corresponds to a member in $\mathbb{R}$. – Brevan Ellefsen Jul 06 '16 at 16:36
  • @EthanBolker However, I have always struggled to follow the line of thinking you use in your answer.... it simply isn't rigorous enough for my thought process. I understand through rigor, and as such I thought I would provide my answer in case that is how the OP best sees it. In actuality we often do just treat functions and mappings on sets... that is the basis of constructing our modern system of $\mathbb{R}$ using set theory, which thus constructs $\mathbb{R}$ as a set and coincides with my answer. – Brevan Ellefsen Jul 06 '16 at 16:38