Capital-Script-D-Of-f

The text kind of recovers from the non-definition I was complaining about yesterday in the next section.

When a function is defined by an equation in x and y, the graph of the function is the graph of the equation, that is, the set of points (x, y) in the xy-plane that satisfies the equation.
Now, that’s a sentence I wouldn’t have to apologize for having written myself (I’d make one change in the punctuation before clicking “Publish”; on the other hand I sort of likexy-plane” as opposed my usual spelling [“(x, y)-plane”]). As I hope to’ve mentioned in all three classes by now, it’s quite the usual thing to “confuse” points-of-the-plane with ordered-pairs-of-numbers. From this point of view, this geometric object right here that I’m thinking about indicating with my index finger (a point) on an imaginary blackboard is not only called “(x,y)”, but actually somehow is the ordered pair (x,y); on this model the Graph and the Function are one and the same.

I sort of had this in mind when I began each class with “Let G be a graph …”. My notes from about Monday night include the equation G := \{ (x, f(x) ) \}_{x\in\Bbb R}; one could actually pronounce this equation “let gee be the graph of eff” … indeed, we could go further and say that f = \{ (x, f(x))\} is true of every function … but this seems sort of weird even to most pros—this is why we don’t go around writing things like (25, 5) \in \sqrt{\null} when we mean 5 = \sqrt{\null}(25) (or more generally (x,y)\in f for y = f(x)). Hmm. Enough weirdness.

The other idea singled out in the summary at the end of this section (3.2 in our text [2.3 in the previous edition]) is the vertical line test, which probably needs no introduction here. The key (14 part) exercise required some review of interval notation, the domain of a function, and the range of a function.

As I mentioned in class, I consider it somewhat weird that no standard notations exist for the last two of these (and that most textbooks have no notations for them at all [except, of course, the Standard English]); I hereby propose the incredibly obvious

{\mathcal D}(f) := \{x| (\exists y) f(x) = y\} and

{\mathcal R}(f) := \{y| (\exists x) f(x) = y\}.

The symbol “\exists” is pronounced “there exists”, by the way …

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  1. ellie

    I was reading ahead for class the other day when I stumbled upon the x-y plane definition. I definitely had to read it over a few times and then thought “wtf”… could they not have written that better?

  2. W1ng5Up

    It appears they did try to write a better definition for the x-y plane. As a survivor of the 4th edition of the Sullivan text, this quarter’s 5th edition is showing some promise. But give credit where credit is due; I suspect any progress I’ve made so far has not been a direct result of the new text edition. . .

  3. the definition of the xy-plane? really?
    these appear to be identical in the 4th & 5th;
    useless for actually *learning* the material,
    i imagine, but maybe a pretty useful *review*.
    (i’m referring here to section 1.1;
    since ellie spoke of “reading ahead”,
    i expect this *isn’t* the passage she meant.)

    my only complaint (a minor one):
    nobody actually says “abcissa” and “ordinate”
    and it’s long past time to quit pretending we do.

    also i’d like to see the symbol {\Bbb R}^2
    in the textbooks in this context … that’s what
    the xy-plane is called in the pros, so why wait?




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