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 ...
- 24.4k
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 ...
- 24.4k
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 ...
- 4,873
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 ...
- 24.4k
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\...
- 10.8k
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 ...
- 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 ...
- 21.2k
23
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.
...
- 581
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 ...
- 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 ...
- 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&...
- 2,721
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 ...
- 17.3k
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 ...
- 3,982
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 ...
- 178
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 $-...
- 1,078
16
votes
Is this just a mistake or a more serious misconception?
It seems clear that there is a certain conceptual gap in the student's understanding. My suspicion is that the student is essentially running the following program in his mind:
Initialize factorial =...
- 2,601
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 ...
- 2,992
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 ...
- 693
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
...
- 5,609
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 ...
- 331
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 ...
- 9,079
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....
- 10.3k
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, ...
- 243
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 ...
- 11.6k
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 ...
- 5,609
13
votes
Is this just a mistake or a more serious misconception?
As per my comment above, it looks to me like the student saw $2k−1$ which is a pretty standard mathematical way to say "odd numbers" and then saw the factorial and thought to themselves "looks like i ...
- 5,030
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), ...
- 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 ...
- 556
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 ...
- 7,520
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