# Why do we teach even and odd functions?

I've been either a student or an instructor in Precalculus or Calculus 1 at about 6 institutions now, and teaching the definition of even functions (where $f(-x) = f(x)$) and odd functions (where $f(-x) = -f(x)$) has been universal.

But why? I don't see how these concepts are so useful that they need to be in the courses that are taught to everyone. I don't see how they lay the stage for understanding calculus.

I mean, seeing how it works graphically is nifty. But it seems like a disproportionate emphasis is placed on these classifications, in every curriculum I've seen.

• it seems like a disproportionate emphasis is placed on these classifications It's extremely useful in applications. What level of emphasis is it that you consider disproportionate? If it's covered in the textbook and mentioned once in a while in examples, then it probably doesn't even need to have 5 minutes dedicated to it in class. – Ben Crowell Sep 11 '17 at 22:36
• I like to teach them because of graphing, integration, understanding the difference between proof by example (not proof) and proof of a general case. (Also: If everyone at your school studies calculus, then you teach at a very different institution than mine.) – Jim H Sep 12 '17 at 0:24
• @BenCrowell It made up a noticeable part of my A-level, with no application at all at the time. – Jessica B Sep 13 '17 at 6:25
• It's an example of symmetry (and anti-symmetry) - which is arguably the most useful concept with a practical application in the whole of mathematics IMO. – alephzero Sep 14 '17 at 1:22
• Many topics in mathematics are not useful. And no one has claimed them to be as well. Again useful means what, can they be of use in future etc are significant questions though. – Sreekanth Karumanaghat Sep 14 '17 at 6:51

One of the major themes of precalculus is what I call “connecting geometry to algebra”. Being able to translate between an algebraic statement like $f(x)= f(-x)$, and the geometric statement that the graph of $f$ is symmetric about the vertical axis is a great instance of this. This is just one more way to practice reinforcing function concepts, and the connection with graphs.

• Even a triple connection geometry - algebra - calculus! – Basj Sep 13 '17 at 20:09

Here you can see that knowing if the function is even or odd can help you when you are integrating over the interval $[-a, a]$.

You can reduce really-hard-to-look-at integrals to zero just by knowing this. As an example, to calculate $E(Z)$ where $Z \sim N(0, 1)$, the standard normal distribution, you have:

$\displaystyle E(z) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}ze^{-z^2/2}dz$

Which is immediately reduced to zero as the inner function in the integral is odd, and you are integrating over $(-\infty, \infty)$.

This may not be too difficult, but knowing this property about odd functions lets you generalize this to all the odd moments of the standard normal: $E(Z), E(Z^3), E(Z^5), ...$ all of them equal zero.

• You could improve this answer by being more explicit here as to why this is useful. eg mention Fourier series – DavidButlerUofA Sep 12 '17 at 6:15
• +1, As an engineer and purely from a practical point this was my first thought. For instance, calculating Fourier series is made much easier knowing about major shortcuts that can be made from the knowledge of odd and even functions. – Lamar Latrell Sep 12 '17 at 15:32

Learning to think about functions abstractly should be one goal in precalculus, and function symmetry helps. Also suppose we carefully protected a student from knowing anything about function symmetry. Upon learning about flux in vector calculus, would this student be able to quickly see that the flux of the vector field ${\bf F}(x,y,z)= y^2{\bf j}$ through the unit sphere is 0?

• Indeed, when I teach multivariable calculus, I show students lots of tricks to simplify integrals. Eliminating terms you know will evaluate to zero after integration, halving the domain so that you can "plug in" zeroes and simplify fewer terms, etc. It all comes down to symmetry. – Matthew Leingang Sep 16 '17 at 13:09

Besides applicability in topics like integration and Fourier analysis, it also connects algebra to calculus at least in the way that multiplication of even/odd functions behaves like addition even/odd numbers:

• Multiplying two even functions gives an even function.
• Multiplying two odd functions gives an even function, too.
• Multiplying an even and an odd function gives an odd function.

Also, you can decompose every function as a sum of an even and an odd function as $$f(x) = \frac{f(x)+f(-x)}2 + \frac{f(x)-f(-x)}2$$ (which is a very useful concept an the same as writing a matrix as the sum of a symmetric and an antisymmetric one as $A = \tfrac12(A+A^T) + \tfrac12(A-A^T)$).

• This also gives one (among many) justification for using the hyperbolic trig functions $\sinh x$ and $\cosh x$ as the odd and even parts respectively of $e^x$ – danimal Sep 12 '17 at 14:08
• +1 Not forgetting that this approach generalizes to order three, four,... symmetries via discrete Fourier transforms and, ultimately, to representation theory of groups. – Jyrki Lahtonen Sep 30 '17 at 6:13

Even and odd parity are probably the simplest examples of function symmetries.

In applied mathematics, the general observation of function symmetries allows to simplify calculations (as stated by others) and to produce more meaningful graphs. In physics, symmetrical parts of a function are sometimes associated to different physical phenomena.

Two examples:

1. If you have a function which is invariant under inversion (a more complex symmetry), that is, $$f(1/x) = f(x),\ x>0,$$ it is better to plot the function by using a logarithmic $x$ axis because $f(a^x)$ is even.
2. The example given by Dirk on the decomposition of a matrix in the sum of a symmetric and antisymmetric part is useful in circuit theory: the symmetric part of the impedance and admittance matrices is associated to the average power dissipated by an electrical network when subjected to sinusoidal excitations.
• This is a better answer than mine. The psychology of voting on these sites is mysterious. – Steven Gubkin Sep 13 '17 at 19:28

I agree that it is disproportionate. Ben Crowell (in comments) says that is helpful for applications. Well, I have a very strong background in natural sciences and engineering, and it is not a big building block. Very few derivations in physics rely on it for instance.

I know this is (-1) controversial, but I think the appeal comes from the ease of doing tricky questions based on it. Sort of ETS style questions. A similar thing would be the vertical angle theorem (or whatever it is called, where a line crosses a couple parallel lines) is big on the SAT.

(-2) I also think that many pure math people prefer things that are definitional and classificational. And that in preference to being able to do multistep mechanical problems like a maximization or power series expansion.

• Really? The difference of Dirichlet versus Neumann boundary conditions (difference between the electromagnetic field behavior near conductors versus insulators) seems to be quite important. The method of images is how a the beginning of electrostatics is understood, and plays a role in the explanation of the Meissner effect in superconductors. The entirety of of the field of crystallography is based on understanding symmetries, and by extension things like understanding diffraction patterns. The difference between even and odd functions also explain the difference in timbre between brass and.. – Willie Wong Sep 13 '17 at 20:53
• ... woodwinds in music, as well as between different types of percussive instruments. And I am merely a mathematician who doesn't do too much natural sciences. I am sure specialists can come up with many more examples. – Willie Wong Sep 13 '17 at 20:56

To express functions as sum of even and odd functions

$$f(x) = f_{even}(x) + f_{odd}(x)$$

And look at the properties of their graphs.

Teach Fourier transform and say it becomes easy to compute when $f(x)$ is an odd function. And the integral becomes zero.

• I think you've missed the point of the question. It isn't about what the syllabus requires, it's about the longer term benefits. – Jessica B Sep 13 '17 at 6:23
• now like applications of it ?@JessicaB – Amruth A Sep 13 '17 at 8:01

Well symmetry helps investigate functions. It's the first things I need to know about the function I deal with. It helps as noted, evaluating integrals that are otherwise difficult to integrate. And it helps visualizing functions. I was a bit disapointed by the few examples given in class ass aplications of these concepts.

As with many items in math that have a name, if we didn't name it, it would still exist.

When manipulating and equation that contained Cos(-x) we are able to observe that this term is the same as Cos(x). On the other hand, when we see Sin(-x) and want to manipulate to form a positive argument, it's equal to -Sin(x).

I don't feel that we spend that much time on the concept. It's introduced in algebra, a parabola possibly being even, a third degree possibly odd, and then again in Trig with the examples I gave. Your experience may be different, more focus than I've observed.

Learning mathematics isn't always about applying things to further mathematics - it can also be about actually applying that knowledge to other areas. So let me bring you a completely different answer to your question, coming from the background of someone that does programming and some web design.

You can easily apply even/odd functions to things like tables: one row has a dark background, the other has a lighter background. A simple enough requirement, that a good programmer might understand not only on the surface, but more deeply such as yourself.

I apologize if this answer is not relevant to the mathematics community directly, but I felt the need (seeing this question in a suggested list) to remind people that math isn't just about math ;)