Skip to main content
42 votes

Why do we teach even and odd functions?

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 ...
Steven Gubkin's user avatar
37 votes
Accepted

Why should or shouldn't we teach functions to 15 year olds?

In the U.S. Common Core standards, functions are supposed to be introduced in the 8th grade, i.e., around age 13-14. So arguably age 15 is a year or two behind where they ought to be. The standard for ...
Daniel R. Collins's user avatar
33 votes
Accepted

What is the proper way to ask a "find the domain" question?

It's not really a question about functions and domains, but about valid expressions. The question is, "For which real numbers $x$ is the following expression valid/well-defined/well-formed?" So, for ...
Daniel Hast's user avatar
  • 4,893
31 votes
Accepted

Why is the concept of injective functions difficult for my students?

I think you will find that almost everyone has this problem when they first starting to learn rigorous mathematics, and many students will never overcome this difficulty. The following three ...
Steven Gubkin's user avatar
27 votes
Accepted

Natural occurrences of a to the (b to the c)?

If your students have learned some statistics, then you could point out that the normal distribution's probability density function uses this double exponential. $$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\...
JRN's user avatar
  • 10.8k
24 votes

How to help new students accept function notation

Start by talking about functions in general, not only about functions that can be expressed by a simple formula in x and y. Examples: The function that maps every non-empty list to its first element. ...
Uwe's user avatar
  • 591
24 votes

How to help new students accept function notation

You might remind them that $y$ is just a name for a number. When they draw a plot, they draw a bunch of points: maybe $y=3$ here, $y=5$ there, and $y=-2$ over there. But at some point (no pun ...
fvy's user avatar
  • 241
24 votes

Examples of relations that are not functions

The example I use: $f(x) = x$'s sister Looks fine, has a formula. I can write "$f($Chris$) = $Jessica" since I have a sister. I can talk about the domain of $f$ by asking for someone to ...
Chris Cunningham's user avatar
23 votes

Why is there a disconnect in the usage of "domain" between high school and higher mathematics, and where does it come from?

In real-world applications, the typical case is that the domain is neither implicit in an expression we write down, nor explicitly stated along with the expression. Rather, one uses knowledge of the ...
user18137's user avatar
  • 231
22 votes

How to help new students accept function notation

You should tell them these two main benefits: (1) Function notation is concise! For example, instead of writing "Find $y$ when $x=5$" one can simply write "Find $f(5)$" This becomes very appreciable ...
Lex_i's user avatar
  • 496
20 votes

Why is the concept of injective functions difficult for my students?

On the other hand, they are really struggling with injective functions. Even after spending a lot of time, they often say "a function is one-one if every element in the domain has a unique image&...
Ilmari Karonen's user avatar
19 votes

What is the proper way to ask a "find the domain" question?

user52817, in the comments, has this exactly right: In the context of Precalculus, the implied domain of a function is the largest subset of $\mathbb{R}$ on which the function is defined. So ...
mweiss's user avatar
  • 17.4k
19 votes
Accepted

What are some examples of great functions that are not too elementary (easy)?

I'm personally a fan of simple examples: $x e^x$ (has nice critical point, point of inflection) $e^{-x^2}$ (with appropriate rescaling, the normal distribution from statistics) $\frac{x}{x^2+1}$ (a ...
TomKern's user avatar
  • 4,425
18 votes

Why do we teach even and odd functions?

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 ...
Dariel Rudt's user avatar
17 votes

Examples of relations that are not functions

An example that should be natural to the students is the square root over the positive real numbers. If one simply says "square root of $4$" then there are two equally nice roots, $2$ and $-...
Martin Argerami's user avatar
16 votes

Why do we teach even and odd functions?

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 ...
Dirk's user avatar
  • 2,991
16 votes

How to help new students accept function notation

The crucial thing the students need to realise is that the (e.g.) $x$ that turns up in the function definition is a bound variable. That's what allows it to be freely renamed or indeed omitted without ...
leftaroundabout's user avatar
16 votes

Why should or shouldn't we teach functions to 15 year olds?

Functions are far broader and more applicable than you give them credit for. Consider the following: Country or state Capital Elevation (in meters) Bolivia Sucre 2783 Ecuador Quito 2763 Colombia ...
Matthew Daly's user avatar
  • 5,619
15 votes

What is the proper way to ask a "find the domain" question?

Although there are good answers already, I think the ambiguity of the vocabulary is being underestimated. This might be because French definitions have more ambiguity than others, I can't tell. In ...
Benoît Kloeckner's user avatar
15 votes

For purposes of teaching, should constant functions be considered "linear functions"?

A linear function is not necessarily a first degree polynomial function: zero function is also linear. In France the terminology is more appropriate than the traditional English one: a linear ...
Alexey's user avatar
  • 331
14 votes

Why do we teach even and odd functions?

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....
user52817's user avatar
  • 11k
14 votes

How to help new students accept function notation

Because x and y are just variable names It happens that sometimes y=f(x), but other times z=f(x,y), w=f(x,y,z), or x=f(y) for that matter. All of these variable names are syntactically equivalent, ...
Monty Harder's user avatar
14 votes

The term "unique" for functions and operations

I don't think there's a lot of educational value to fixating on trying to word definitions in exactly the perfect way. Students have trouble with the notion of a function because it's hard. The way ...
Henry Towsner's user avatar
14 votes

Why do we use functional composition in the order we do?

In general applications, defining the evaluation rule of $f\circ g$ by $(f\circ g)(x)=f(g(x))$ has a lower extraneous cognitive load than the way you suggest. As a sample of how inconveniencing that ...
Matthew Daly's user avatar
  • 5,619
13 votes

How to help new students accept function notation

TL;DR: A function is a verb. It's an action. Variables are nouns, objects. Verbs (functions) connect nouns semantically, i.e. how A (or x) relates to B (or y), ...
ΦDev's user avatar
  • 131
13 votes

Why is there a disconnect in the usage of "domain" between high school and higher mathematics, and where does it come from?

We give high-schoolers many different explanations of the word "function." Here are a few that are either implied or outright stated at various points in a student's education: A function ...
Kevin's user avatar
  • 586
13 votes
Accepted

Why do we use functional composition in the order we do?

The answer to the lede question "Why do we use function composition?" is "Because we do." It is the way that, for historical reasons, the notation developed. It is a convention ...
Xander Henderson's user avatar
  • 8,225
13 votes

Examples of relations that are not functions

Given points $P = \{x_1,x_2,\ldots \}$ on a line $L$, let $nn(x)$ for $x \in L$ be the nearest neighbor to $x$: the point in $P$ closest to $x$. For most $x$, there is a unique $nn(x)$, but if $x$ is ...
Joseph O'Rourke's user avatar
12 votes

For purposes of teaching, should constant functions be considered "linear functions"?

While I haven't done a systematic survey, my impression is that the overwhelming majority of pre-calculus and calculus texts define a linear function to be one of the form $f(x) = mx + b$ with no ...
John Coleman's user avatar
  • 1,536

Only top scored, non community-wiki answers of a minimum length are eligible