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 ...
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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 ...
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32 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 ...
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  • 4,765
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 ...
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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\...
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  • 10.2k
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 ...
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  • 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 ...
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  • 19.1k
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. ...
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  • 580
22 votes

Should we teach functions as sets of ordered pairs?

I wanted to provide a different perspective on this. I just recently graduated with an undergraduate degree in mathematics, and I wanted to say a few things about what personally helped me in ...
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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 ...
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  • 496
22 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 ...
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  • 221
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&...
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19 votes

Should we teach functions as sets of ordered pairs?

In my experience, the "set of ordered pairs" is a difficult and confusing definition. Moreover, I feel it's almost totally unnecessary because in practice in mathematics we do always treat functions ...
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  • 6,238
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 ...
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17 votes

Should we teach functions as sets of ordered pairs?

Let me give you yet another point of view which is a bit closer to discrete mathematics, that is, from introduction to functional programming. As this course is very function-heavy, it was worth to ...
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  • 8,729
17 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 =...
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17 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 ...
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  • 16.3k
17 votes

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 ...
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  • 1,939
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 $-...
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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 ...
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  • 2,932
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 ...
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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 ...
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  • 5,539
15 votes
Accepted

Why are hyperbolic functions given "short shrift" at "low" levels of math?

There are a few reasons I can think of, not all of them strictly pedagogical. The trigonometric ratios (as opposed to functions) have a direct relationship to measurement of triangles, which are an ...
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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 ...
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  • 271
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....
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  • 7,585
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, ...
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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 ...
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  • 5,539
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 ...
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  • 4,840
13 votes
Accepted

How can we focus students on the various data types in multivariable calculus?

I might seem picky, but I would first refrain from saying that $\nabla f$ is a vector. It is a vector field. This might be considered a common abuse of vocabulary, but using it amounts to assuming ...
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