# Tag Info

### 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 ...
• 19.5k
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 ...
• 20.9k
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 ...
• 4,765
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 ...
• 19.5k
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\...
• 10.2k

### 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

### 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 ...
• 19.1k

### 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. ...
• 580

### 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 ...
• 329

### 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

### 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 ...
• 221

### 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&...
• 2,602

### 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 ...
• 6,238

### 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

### 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 ...
• 8,729

### 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 =...
• 2,513

### 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 ...
• 16.3k
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 ...