Math

The Physics Philes, lesson 19: How Do You Function without Math?

In which functions are solved, graphs are drawn, and fun is had.

I’ve always had a heaping ton of math anxiety. Ever since I was very young, before I knew that girls weren’t supposed to like or be good at math. I consider myself a relatively intelligent person. I don’t know why numbers scared me so much. But they did. Consequently, I didn’t have very many “ah ha!” moments.

One such moment came when I was in grade school. We were learning how to estimate. I don’t remember exactly what aspect of estimation was giving me trouble, but for some reason I couldn’t get the hang of it. Until my dad asked me to count how many walnuts were in a bucket. If I remember correctly, there were 58. But that’s not what I said. I said that there were about 60. Then, out of nowhere, my mom yelled out the window, “Mindy! You estimated!”

Holy crap. I did estimate! Yeah, it’s not an impressive thing to brag about now, but at the time I was on cloud nine. I finally got it.

I tell you this dull and inane story because, since I began studying math again in a systematic way, I’m having more and more of those “ah-ha!” moments. I don’t really know what my problem was the first time around. Maybe I’m just ready to learn it in a way I wasn’t in high school. At the very least, maybe it will help me delay the onset of dementia, so it’s really a win-win.

Now, where did I leave off? Oh yes. Functions.

A function is a special kind of relation. A relation in math is basically any sort of mathematical equation. What makes a function stand out is that we know how it will behave. You see, if an equation is a function, no input is allowed to give more than one output. For example:

Look at the first equation. If we decide that x = 2, the output will always be 4. So the first equation is a function. However, that’s not true of the second equation. In that relation, if we decide that x = 2, the output could be 4 or 0. Since it’s possible to get more than one output, the second relation is not a function.

Let’s take a step back for a moment. What are these inputs and outputs I keep talking about? Perhaps you’ll recognize the concepts by their other names: domain and range. A domain is the set of inputs for a function. Any number a function accepts as an appropriate input is part of the domain. A range is the set of outputs given by a function.

Say we have a relation s: {(-1, 5), (1, 6), (2, 4)}. This is basically a list of inputs and outputs. The domain of this relation is {-1, 1, 2}. The range is {4, 5, 6}.

Now let’s incept this concept. We can plug a function into another function. It’s called composition of functions, and it’s easier than it sounds. I’ll show you.

If f(x) = √x and g(x) = x + 6, evaluate g(f(25)).

You want to start from the inside out. Regardless of how many functions within a function you have, just start with the innermost function and work your way out. We start by plugging in 25 to the f function.

f(25) = √25 = 5

Next, we’ll plug 5 into the g function.

g(5) = 5 + 6 = 11

So the function g(f(25)) equals 11. See? I told you it was easy!

This week I ran across something kind of weird: the piecewise-defined function. This type of function looks like this:

What this means is that if the input is a number less than 2, plug it into the top equation. If the input is greater than or equal to 2 you need to plug it into the bottom equation. It looks complicated, but it’s not.

Functions are much easier to deal with than I remember. But now we need to figure out how to tell if a function is symmetric. A function has symmetry if it mirrors itself with respect to a fixed part of the coordinate plane.

That is a horrible sentence. But it becomes more clear as we get into the specific kinds of symmetry.

When a graph looks the same on either side of the y-axis, that graph is y-symmetric.

This graph is y-symmetric.

Luckily, you don’t have to actually graph the equation to tell if the graph is y-symmetric. There is a test. All you have to do is replace all the x’s with (-x) and simplify. Let’s take the equation of the graph above and put it to the test.

Just replace each x with a (-x).

Simplified, the (-x) equation is identical to the original. That means the the graph is y-symmetric. If there had been a sign change or anything different, that is a sign the graph is not y-symmetric.

There are other types of symmetry, too. If a graph is identical above and below the x-axis, the graph is x-symmetric. The test for x-symmetry is almost identical to the test for y-symmetry, except that instead of replacing all the x’s with (-x), you replace all the y’s with (-y). A graph is origin-symmetric if the graph does exactly the opposite thing on the other side of the origin.

This graph is origin-symmetric.

The test to determine if a graph is origin-symmetric is to replace all the x’s with (-x) and all the y’s with (-y). If the simplified equation matches the original, the graph has origin symmetry, just like the others.

So, those are functions. But did you know there is such a thing as an inverse function? I was a little scared when I found that out, but once I got into it, even those aren’t that bad.

An inverse function is a tool you can use to eliminate something unwanted from an equation. Two functions are the inverse of each other if you can plug them into each other and only x remains. Or, in math:

f(g(x)) = g(f(x)) = x

We know how to do this thanks to composition of functions. But inverse functions don’t just spring out of thin air (usually). We can make them! Just follow these simple steps.

(Note: That little negative 1 there? That’s how we indicate an inverse function.)

First, replace g(x) with y:

y = √2x + 5

Next, switch the x and y. This turns the function inside out and cancels out the original equation as a result:

x = √2y + 5

Solve for y:

Finally, write the answer in proper inverse function notation:

Ta da! We have the inverse function of g(x) = √2x + 5.

So now we have function basics. But what if a graph isn’t a function? What happens then? Don’t worry. The world won’t stop turning. We can use parametric equations in those circumstances. Parametric equations are pairs of equations that are used to express non-function graphs. You’ll usually see them as “x =” and “y =” equations. The graph is defined in terms of a third variable t, or parameter. A parameter is a variable into which you plug in a number to find the coordinates.

For example:

x = t – 1

y = t – 2

All you have to do is plug in any value for t and get one coordinate at a time. But to get a graph this way would take forever. Luckily, all we have to do is translate the parametric equations into rectangular form. Don’t panic. I’ll show you how. I’ll translate parametric equations x = t + 1 and y = t – 2 into rectangular form.

Step 1: Solve on of the equations for t. I’ll solve the x equation, but you could use either one.

x – 1 = t + 1 – 1

x – 1 = t

Step 2: Now that t is in terms of x, replace t in the y equation

y = t – 2

y = (x-1) – 2

y = x – 3

The result of this process is an equation in slope-intercept form! We’ve already been over what to do with those!

Whew! Lots of functions; lots of graphs. And, unexpectedly, lots of fun. Let me know if I’ve misunderstood something, or if I got anything completely wrong. I learn a lot from your comments.

Featured image credit: Evelyn Saenez

1. It doesn’t have to be parametric. The graph for a circle, which is not a function, is x^2 + y^2 = r^2

2. A couple of comments on the language around functions.

First off, to a mathematician, the type of a function is just as important, and sometimes even more important, than its formula.

Consider these two functions. You may or may not remember that Z is the set of integers (whole numbers, including negative ones) and R is the set of real numbers.

f : Z → Z
f(x) = x²

g : R → R
g(x) = x²

These functions have the same formula, but they are different functions to a mathematician. So I’m always going to give the type of a function unless it’s clear what the type is. Sorry if that’s a bit verbose, but it took me a while to get into this habit, and I’m not breaking it for you.

Most high school students know that if the type of a function is:

f : A → B

then the set A is called the “domain” and B is called the “codomain”. Why “codomain”? Well, it turns out that it’s one of a bunch of pairs of “X”/”co-X” words from a branch of maths called category theory. Category theory is very abstract maths, and most people will never encounter it. But I figured I should mention it because it always puzzled me back in the day.

The codomain of a function is different from the range of a function. The range always a subset of the codomain; sometimes it’s a proper subset and sometimes it isn’t. The codomain is a property of the definition of a function, but the range is a property of its behaviour.

Suppose we have a function f : A → B. An inverse of f is a function g : B → A such that f(g(x)) = x and g(f(y)) = y. Note that it’s not necessarily correct to say:

f(g(x)) = g(f(x)) = x

This is only correct if the domain and codomain of f are the same. This is sometimes important, even in physics, so please be clear on this.

If f has an inverse, then there are at least three different terms for this:

1. f is invertible, or f has an inverse.
2. f is a bijection, or f is bijective. (This is the “set theory” term for it.)
3. f is an isomorphism, or f is iso. (This is the “category theory” term for it.)

The three terms are actually saying implying slightly different things, but the technical distinctions are not important right now. But nonetheless, you might hear words like “bijection” or “isomorphism”; they just mean “invertible”.

Finally, it’s worth pointing out that there are other kinds of inverse.

Consider the functions:

s : Z -> Z
s(x) = 2x

r : Z -> Z
r(x) = x/2, if x is even
r(x) = 0, otherwise

Then r(s(x)) = x, but s(r(x)) is not always x. We call r a left-inverse of s, and s a right-inverse of r.

The attentive reader will note that we could have defined r to return anything if x is odd and it would still be a left-inverse of s. That’s why we speak of “a” left-inverse rather than “the” left-inverse.

Incidentally, I named them r and s because they also have category theory names which you are unlikely to see even at the undergraduate level, but nonethless here they are. We call r a “retraction” of s (you can think of it “undoing” or “retracting” whatever s does). We call s a “section” of r (think “cross-section”). Be the life of nerdy parties by dropping these terms into conversation.

OK, now back to symmetry.

What is here called a “y-symmetric function” is, to most mathematicans, called an “even function”, and an “origin-symmetric function” is called an “odd function”.

Why “even” and “odd”? Well, consider a polynomial with only even powers in it. Then that function is even.

f : Z → Z
f(x) = 2x² + 1

Remember, 1 is x to the power of zero, and zero is even.

Similarly, a polynomial with only odd powers in it is odd:

f : Z → Z
f(x) = 3x³ + 4x

Again, x is x to the power of 1, and 1 is odd.

Even and odd functions seem like a useful concept to keep around because of the symmetry, but it doesn’t seem like they are that generally applicable to the functions you might need to use.

However, any function (of the right type) can be decomposed into even and odd parts! Suppose you have any old function:

f : R → R

Now consider the two functions:

f+ : R → R
where f+(x) = (f(x) + f(-x)) / 2

f- : R → R
where f-(x) = (f(x) – f(-x)) / 2

You can verify for yourself that f+ must be even, f- must be odd, and f+(x) + f-(x) = f(x). Neat, eh?