Why do we teach even and odd functions?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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 $A = \tfrac12(A+A^T) + \tfrac12(A-A^T)$).

Answer 4

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?

Answer 5

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.

Answer 6

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.

Answer 7

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.

Answer 8

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.

Answer 9

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.

Answer 10

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 ;)