y = x vs. f(x) = x

[am77494]

From my college, engineering notation :

y = sqrt (x) means the positive or the negative square root of x. This is this curve

This is a very valid representation of y as it relates to x and is used in algebra.

For calculus

f(x) = sqrt(x) means only the positive root of x. Because in calculus functions cannot be one to many mappings. There must one and only one value of f(x) for every x.

Here is another example

y = arcsin(x) is valid in trigonometry; f(x) = arcsin(x) needs to be qualified for f(x) domain to avoid multiple values of f(x) for same x

[Tim_R.Mortiss]

@friedo

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.

Yes, they are specific examples of valid but trivial functions. But I don’t think that’s what the OP meant.

[GargoyleWB]

@Great_Antibob

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.

Even worse when combined with the typical notation of i j k vectors in calculus. My physics prof set a ground rule early on that mathematicians and EEs should be exiled to a small island for their crimes against notation, but he may have been a bit biased.

[Acsenray]

@Tim_R.Mortiss

Yes, they are specific examples of valid but trivial functions. But I don’t think that’s what the OP meant.

I think it is what he meant, at least in part. I think he wants to know why the notation form “f(x)=” exists AT ALL, regardless of what is to the right of the equals sign. Why not just use “y=”?

[Thudlow_Boink]

@Crafter_Man

I’ve never been a fan of this notation, since it looks like “f multiplied by x equals x.”

@Musicat

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.

Whether you like it or not, f(x)-style functional notation is so firmly entrenched and so convenient that it’s not going anywhere.

[D18]

@Acsenray

I think it is what he meant, at least in part. I think he wants to know why the notation form “f(x)=” exists AT ALL, regardless of what is to the right of the equals sign. Why not just use “y=”?

Yes, this is what I was getting at. I didn’t know if there was a meaningful difference between the two, or if they were essentially mathematical synonyms.

[Leo_Bloom]

@friedo

“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.

+1

[The_Hamster_King]

@Chronos

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”.

This. Function notation is a handy abstraction for more complicated math.

For example, right now I’m playing around with Bezier curves. The equation for one of these curves is V = f(t) where t is a control parameter on the interval [0,1], V is a two-dimensional vector representing a point on the curve, and f(t) is a complicated function involving multiplying coefficients by powers of t and summing them. It would be a pain in the ass to have to write out the full expression of f(t) all the time. It’s much nicer to be able to write something like

W = f(t) + g(u)

when you’re trying to work through a problem.

[Pedro]

Beyond issues of notation, the point is that an algebraic function always defines an equation or system of equations, while the reverse is not true.

In the special cases that it is true the notation is interchangeable, out of habit and practicality.