While seeing some math stuff I realized I'm confused with some of the notation to distinguish functions and equations. I have some points where I have some doubts and would like to clear them out, would be grateful if someone could answer to all of them.
1)
Let's say we see this written somewhere:
$$y = x +1$$
Should I see this "thing" as an equation, a function, or does it depend on the context? Here is my confusion:
I would read that as "a value named '$y$' is equal to a value named '$x$' plus $1$", so an equation with two unknown values where the solutions are pairs $(x,y)$ that satisfy the relation (equation). I don't see it as a function because I feel that the "thing" is a problem-like thing which expects a solution; a function on the other hand doesn't have solutions, is just a function. For me a function would be like this:
$$y(x) = x +1$$
Since now I can feel that '$y$' requires an '$x$' to give a certain output.
2)
What about when I want to use a function inside an equation? Like saying that a function $y$ minus $1$ is equal to a value named $x$, which notation should I use?
A: $y(x) – 1 = x $
B: $y – 1 = x$
In 'A' I understand better that '$y$' is a function even though it gives me the feeling that it isn't really the function that is subtracting by $1$, but instead its value when given an $x$, but maybe that does mean the same (?).
In 'B', without context I would see '$y$' as a unknown value, not a function.
But my confusion is: if 'A' is an equation, it has solutions right? And those solutions I think they would be like a pair $(x,y)$, where $x$ is a value and $y$ is a function.. does that make sense?
But as soon as I rearrange the equation to this: $y(x) = x + 1$, is it now suddenly a function (something that doesn't have solutions), or is it still an equation (something that has solutions)?
3)
Those two points above make me infer a weird thing, is this possible?:
$$y(x) = y$$
As in "a function $y$ is equal to a unknown value named $y$"?
4)
If someone says before-hand that $y$ is a function, and then writes this:
$$y = x + 1$$
Is he wrong for not using the "$y(x)$" notation?
Best Answer
Short answer. You are right to be confused.
The meaning of the equation $$ y = x + 1 $$ does indeed depend on the context. You can't understand it if you just
If it's meant to be a function it might or should be named $f$, or some such. Then you would see $$ f(x) = x + 1 $$ or perhaps $$ y = f(x) = x + 1. $$ I've rarely encountered $$ y(x) = x+ 1 \ . $$ If I did I would grumble but would know what was meant.
Just that equation in some other context might be the equation of a line in the plane.
Related, possibly helpful: What exactly is an equation?
Edit, in response to a comment asking
Think about how to interpret an equation like the familiar $$ x^2 + y^2 = 1 . $$
The context will probably tell you that's meant to specify the set of points $(x,y)$ whose coordinates satisfy that equation - the unit circle in the plane.
Sometimes you may want to think of the equation as determining a value of $y$ given a value of $x$ - that is, think of $y$ as a function of $x$. In this case each value of $x$ between $-1$ and $1$ determines a unique positive $$ y = \sqrt{1 - x^2}. $$
Often you won't be able to "solve for $y$" explicitly. For example, the points that satisfy $$ x = y^5 + y $$ do define $y$ implicitly as a function of $x$, but there's no formula that tells you how to calculate it.