## We Bring it on Ourselves

$\exists \,\in$ confusion.

$f(g(x))\Leftrightarrow (f\circ g)(x)$, but don’t confuse that with $(f\cdot g)(x)$

$f(g(x))\neq g(f(x))$, unless of course you’re trying to show that a function is the inverse.

Speaking of which…

${f}^{-1}(x)\neq \frac{1}{f(x)}$, but ${x}^{-1}= \frac{1}{x}$

Let’s not even get into onto.

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### 17 Responses to We Bring it on Ourselves

1. Ben Chun says:

Inflammable means flammable? What a country!

2. Jackie says:

Hi Dr. Nick!

Gotta love the student aide who, among other things, catches the references made on your blog. Thanks Andy.

3. This is so funny. I’ve never really thought about it, but it is quite confusing when you put all together. Good post.

4. Jackie says:

Thanks. It’s more amusing here than it is in the classroom though. Now I can laugh.

5. jd2718 says:

x comes before y, but slope and tan are y/x. Sine looks basic, cosine derived. Then how come cosine goes with x and sine with y?

And I get, but the students often don’t, why k moves f(x) + k in the y direction k units, but h moves f(x + h) in the x direction -h units.

6. vlorbik says:

$sin^{n}(x) = [sin(x)]^n$
when $n \in \{2, 3, 4\} \ldots$,
but means something else when $n=-1$
(and goes unused altogether for other values).
this is a doggone shame. surely we oughta have
$f^{\leftarrow}$ or something to denote inverses.

the rest don’t bother me.
as for $\in$ and $\exists$ you might as well complain
that it’s easy for beginners to confuse d with b
(this is presumably why earlier generations were taught
to “mind their p’s and q’s”; i expect this is passing
from the language pretty quickly though …).
and $f\cdot g$ should be rare-to-nonexistent
in our context (“precalculus”, say).
that things not known to commute
must not be assumed to commute
isn’t something we bring on ourselves at all;
it’s a fact of nature. if students find it confusing,
well, that’s job security for us. likewise for the
contravariant behavior associated, as in jonathan’s
example, with adding before applying a function.

could be i’m wrong of course … got a better idea?

7. vlorbik says:

oh, and “onto”. why not get into it?
my opinion? — thanks for asking.
the real problem is with
the usually-ill-defined term “range”
which should be split forthwith
into “target” and “image”.
thus, given that $f$ satifies
$f : \mathbb{R} \rightarrow \mathbb{R}$, we can say that
$f$ has the full set of real numbers as its “target”
but if moreover, we know that $f = [ x\rightarrow x^2]$,
then $f$ has $\null[0, \infty)$ as its “image”
(and the question of “onto” simply doesn’t arise).

8. Jackie says:

Do I have a better idea? Nope. Job security? Yep.

I like the idea of splitting “range” into “target” and “image” though, thanks.

9. jd2718 says:

Vlorbik,

somewhere along the way I picked up, and held this usage:

Domain, as usually defined, Codomain (your target) and Range (your image). No matter what the right way of naming it is, it is a distinction worth making.

As to f(x) + k vs f(x + h), I had some luck last time working with the graphs of

y – k = A*funct(x – h) (absolute value first, since it is unfamiliar and easy)

(working off the observation that for the equation of a circle centered at (h,k), the kids seem to get the minus signs right pretty easily)

$(x - h)^2 + (y - k)^2 = r^2$

10. Beans says:

(Whoops could you please remove the comment above! I am not with it today. :o) [Done! I fixed the LaTex too. JB]

I have never got confused with f(g(x)), but I think confusion about ${f}^{-1}(x)$ arises, because I seem to recall something about dy/dx= $\frac{1}{dx/dy}$. (Or something similar?!)

Vlorbok: About the onto business – I definitely agree! I will re-read what you wrote (whilst I am fully alert) because something tells me it will make sense. Because of this onto business, I hate surjective functions and the word onto. (Memories of my linear algebra exams and kernels spring to mind). GAH. (Finally a place where I can express my dislike for them!)

11. Vlorbik is even more right because the distinction between “target” and “image” is what real mathematicians do. In fact, when you generalize away from “functions” between “sets”, everything has a target, but we can’t always even define what an image is.

On the other hand, there’s a big schism between your seemingly more sensible definition of composition and the more traditional way. The usual way says that $f\circ g$ means “first do $f$, then do $g$ to get $g(f(x))$“.

12. beans says:

Thanks for the edit and the removal. (It’s pretty cool the way you can edit comments!)

I think with composition it depends on who taught you. One lecture I went to, (f o g)(x) was defined to be g(f(x)) whereas I define it to be f(g(x)).

13. Jackie says:

John – We’re not quite into set theory yet. I’m keeping in mind defining using target and image (why not set them up for a bit of “real” math?).

John & beans – I’m a bit perplexed by the different interpretations of composition. I wish I had time at the moment to look a few things up. Alas, the outside world calls, I’ll be back in a few hours.

14. Oh, I didn’t mean you’d say it in those terms. Still, it’s just as easy to say “target” and “image” instead of “range”, and closer to what real mathematicians say. And if the kids do eventually get to that level, so much the better.

15. Jackie says:

I’m still perplexed by this: “The usual way says that $f\circ g$ means “first do $f$, then do $g$ to get $g(f(x))$“.”

I’ve always understood it to mean first do $g$, then do $f$ to get $f(g(x))$.

MathWorld seems to support this too.

Also, I agree that it’s just as easy to say “target” and “image”. I’ll be bringing it in to our conversations – but probably not the week before finals (I don’t want to confuse them). This will have to wait until second semester.

16. It’s really about 2/3-1/3, in my experience. John Baez makes specific note that he breaks with the standard in his quantum gravity lecture notes to write it your way, as well as in most of his higher-dimensional algebra papers. Most abstract algebra textbooks I’ve seen write with the convention I stated, though the Wikipedia entry on function composition does it your way.

Really it doesn’t matter, as long as whoever’s talking or writing is clear about which order of composition is intended. It also modifies things like whether operations are covariant or contravariant, and it swaps the definitions of left and right modules over a ring. If only mathematics had a body like l’Académie française to just settle the question and impose a standard.

17. Jackie says:

Thanks for the link and the clarification.