Confused with the notation of functions and equations

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?

Solution 1:

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

see this written somewhere.

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

What about when I want to use a function inside an equation?

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.

Solution 2:

Fundamentally, an equation is just a segment in a sentence, or sometimes a full sentence. For example, if I write:

Let $x$ be a number such that $x^2 = 4$, then $x < 4$.

This is a sentence, where I replaced some of the words with symbols. Of course, you could write the same using words:

Let $x$ be a number such that $x$ squared is four, then $x$ is less than four.

So when you see $x^2 = 4$, you should just think of it as the words “$x$ squared is four”. Is that a problem you can try to solve? Well not really. Is that a function? Well not really either. It's just a statement, like if I said “My car is blue”. In fact it doesn't even have to be true! I can say $1 = 2$ without any issue. It's a lie, but I can still say it, just like I could say my car is blue even though I don't own a car.

Depending on the surrounding sentence, it may become a problem, e.g. if you see “Find a number $x$ such that $x^2 = 4$”, then you have a problem you can try to solve! But the equation is not the problem — it requires the surrounding sentence.

Now, you might be confused because your school assigns you problems just by writing an equation. But such a problem would typically be associated with the words “solve the equation ...”, which is short-hand for “Find all possible numbers $x$ such that ...”.

As an example “Solve the equation $x^2 = 4$” is short-hand for

Find all numbers $x$ such that $x^2 = 4$.

Notice in particular that the equation really is part of the sentence; in fact the equality symbol is the verb!

Now, let's talk about $y = x + 1$ vs $y(x) = x + 1$.

You say that a function is something like $y(x) = x + 1$, but this is a misconception. It's just a statement about some things denoted by $x$ and $y$. Take the following example

Let $y$ be the function given by $y(t) = 2^t$, then we have $y(x) = x + 1$ when $x = 1$.

Notice how this sentence contains $y(x) = x + 1$, but the function $y$ is not at all the function that adds one. It's just a statement about $y$ and $x$, which in this case happens to be true. (because $2^1 = 1 + 1$)

Additionally, the equation $y(x) = x + 1$ is not the function. The function is just the thing we denote by $y$, and the equation is a statement about this function that gives us some information about it.

You say this:

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

Your feeling that $y(x)$ is the value when given $x$ is exactly correct! When I say $y(x)$ I am in fact talking about some number. If $x$ is another unknown number, I may not know which number $y(x)$ is, but I assure you that it is a number. If I want to talk about the function, I just say $y$.

Note that most people, including many teachers do not appreciate that there is an difference between $y(x)$ and $y$, but I find it unlikely you will get into trouble by doing it correctly.

Is $y(x) = y$ possible? Generally this is misuse of notation. The left side is a number and the right side is a function.

However you must unfortunately be prepared for people to misuse notation.

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?

Well, if $x$ is a number then yes, although I would like to introduce something you may find interesting: if $x$ was another function, then it could indeed be perfectly valid.

The idea is that you can think of think of functions as values you can manipulate just like you can with numbers. You are most likely to encounter this concept in the form of the function composition operator, which is written like this: $f \circ g$.

Basically what it means is: Given two functions $f$ and $g$, the expression $f \circ g$ represents a third function such that $(f \circ g)(x) = f(g(x))$ for every $x$. So this means it's treating functions like something you can do something with and that you can build new values by putting others together, just like $+$ does to two numbers. (Note that people can't agree whether it should be $f(g(x))$ or $g(f(x))$.)

Now, it does make sense to ask if you can add two functions. It turns out that you can define this in a way that makes sense, namely if $f$ and $g$ are functions, then $f + g$ is a function such that given any number $x$ we will have $(f+g)(x) = f(x) + g(x)$.

I hope this helps you build some intuition for what equations and functions are, and feel free to ask any questions.

Solution 3:

1) Functions are defined in term of relations i.e a relation $\rho$ is a relation if for all $x$ there is an unique $y$ such that $x \rho y$ but $x\rho y$ is just a shorthand for writing $(x, y) \in \rho$.

Now this is very useful because we graph functions i.e $y=x+1$ by using that $(x, x+1)$ is in relation which gives as a way to describe both functions and relations.

We can graph something as $3y+2x+9=0$ and this describes a function but $3y(x)+2x+9$ can be misunderstood as $3yx+2x+9$ and if you have multiple occurances of $y$ the notation can become a little messy. We can even draw something as $x^2+y^2=1$ (circle) and $x=y^2$(parabola); in both $y$ isn't a function of $x$.

I feel it's nice intuition that connects two seamingly difderent concepts i.e equations and functions/relations so we can describe a solution set (of a equation) graphically.