Is there a practical and/or philosophical difference between y=x and f(x)=x, or are they just two different notations of the same thing?
Not homework - just trying to revisit some basic math concepts in attempt to get my head around them.
They both mean pretty much the same thing. The first has the additional meaning that the function of x I’m talking about I will call y. The second has the additional meaning that I am labeling the function f.
There’s a technical difference that most people probably won’t need to know for practical purposes but becomes important if you want to do anything more than minimally simple with it.
The second notation, i.e. f(x) = x, means that you have a “function” which is named f. Further, it’s a function of x, letting you know the independent variable here.
The first notation is generally the same as the second, except it’s implicit. We take y to be a function of x, i.e. y = y(x). In this case, the value of x tells us the value of y.
Of course, it’s possible (but not usually the case) that we actually just have an equation in two different variables x and y. Context can usually tell us if this is the case or not. An equation is of course different from a function.
To expand on the last point, we might have a system of equations like:
y=x
y^2 = x
y+3y = 3x^3
In this case, y=x is simply an equation and not a function.
Many pedantic people consider f(x)=x an abuse of notation.
The “correct” notation is
f: x ↦ x
This isn’t very clear without a more complex function :p. Let’s try y = x+2
f: x ↦ x+2
This is read “f is a function which maps x to x+2.”
Then, y=f(x). The notation “f(x)” is “supposed” to always denote a value, so “y=f(x)” reads “y is the value of f evaluated at the value x.” Thus “y = f(x) = x+2”.
However, in common terminology, including your high school algebra class probably, they’re pretty much interchangeable because the difference isn’t extremely interesting or useful in those contexts. In higher levels of math, the distinction can be necessary, but even then this “abuse of notation” isn’t really that uncommon.
Edit: Great Antibob has a point that the distinction becomes more noticeable when you have systems of linear equations, I didn’t think about that.
Are you asking about the identity function f(x)=x in particular, or does your question apply equally to other functions, like y = 3x+5 vs f(x) = 3x+5?
Assuming the latter: for most practical purposes, they are basically just different notations for the same thing, although that means that one notation may be more natural or more convenient than the other in certain contexts. For example, instead of saying “What is the value of y when x is 17?” you could just say “What is f(17)?”
Thank you, every one.
Yes, I meant in general, with the equation in the OP as an example.
I’ve never been a fan of this notation, since it looks like “f multiplied by x equals x.”
In practice, I don’t distinguish between them. Even more correct is f = \lambda x x. If that sort of thing turns you on, be my guest.
“More correct” by what metric? We do not generally expect high school algebra students to do lambda analysis of functions. There is nothing “less correct” about f(x) or f: x ↦ … notation, it just depends on what level of formality is relevant to the problem at hand.
I can draw two blobs with arrows connecting things inside them and that’s a perfectly valid and correct notation for a function, too.
y = √x
is not a function (though it’s a relation), so would it be nonsensical to say
f(x) = √x
?
y = √x is a function of x if the radical symbol is taken to mean the principal (i.e. positive) square root, which is generally the case in algebra. That’s why your teacher made you write ±√x when solving quadratic equations.
So writing f(x) = √x is perfectly fine.
Oy
That always bothered me, too. In 8th grade beginning algebra, we were taught that a(b)=c meant a times b equals c. Substitute letters at random, and x(y) = z is essentially the same or x times y equals z. Use any letters you want, anytime.
Then we find out that f(y) = z doesn’t mean what it appears to mean, but that’s only because mathematicians recognize “f(” as a special character. Which isn’t all that problematical when you learn that not all characters are ambiguous after all. i, or e, for example.
Not the best example, actually. Electrical engineers use i to represent electrical current, so they use j to represent the imaginary unit. It made for some interesting mistakes when I took math and EE classes in the same semester in college.
Actually, as noted, the f(x) notation is just one notation. There are other ways of expressing it. This one just happens to be favored. Notationally, there are few, if any, things in math that are set in absolute stone.
I think a distinction can be made between “i” (italic), meaning the square root of minus 1, and “I” (upper case), meaning current, as in E=IR. At least that’s the way I used to use both characters.
You can do it that way, but the closest thing to a standard notation in electrical engineering is j for the imaginary unit for disambiguation purposes.
You are, of course, welcome to use whatever symbols make you comfortable, but that’s kind of my point. The closest thing to a standard is still a bit ambiguous, and it all comes down to convention in the end. And, in the case of the OP, sometimes dealing with notation and representation gets in the way of understanding. Unfortunate, but I don’t see a realistic way of getting around it.
“The nice thing about standards is that you have so many to choose from.”
–Andrew S. Tanenbaum
They are both wrong. The proper notation is:
y = f(x), where f(x) = some formula based on x, such as f(x)=5x^2 + 17x + 11
x is the independent variable, y is the dependent variable, and f is the function or formula that finds the y which corresponds to any given x.
To say y=x is flat out untrue, as is f(x)=x.
An important distinction, here: Are you interested in the values of the function at various points, or are you interested in the function itself? A statement like “y = x+2” works just fine for the values, but doesn’t give you a name for the function. On the other hand, if you give the function itself a name, such as “f”, or “g”, or “sine”, then you can make statements like “the derivative of f is g”, or “the derivative of sine is cosine”.
y=x and f(x)=x are both perfectly good function definitions. There is no rule saying a function can’t map values in its domain to themselves.