Is $x$ in $ax+b$ every value, or a specific unspecified value?

Question

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?

Answer 1

$x$ 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 $y$.

Answer 2

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.

Answer 3

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 $f(x)$