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Questions tagged [functions]

For questions about the teaching of the function concept, function properties, and various types of functions.

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25 votes
5 answers

"Function" vs "Function of ...": how much does it contribute to students difficulties?

Most textbooks I've seen (and teachers I've met, myself included) are rather careless about the distinction between variables and functions. For example, when we write $y=f(x)$ we all know that $f$ ...
Michael Bächtold's user avatar
24 votes
6 answers

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

A function is not really a function unless it's defined everywhere on its domain. So consider these three questions: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the square root function $f(x) = \...
Paul Castle's user avatar
8 votes
3 answers

What are some common ways students get confused about finding an inverse of a function?

What are some common ways students get confused about finding an inverse of a function? One I can think of is conflating multiplicative inverses of rational numbers with functional inverses. e.g. ...
Eleanor Hankins's user avatar
45 votes
21 answers

How to help new students accept function notation

I am struggling to help some of my new precalculus students accept function notation -- something new to them this term. I am looking for strategies to help them adopt this new notation. Their main ...
Nick C's user avatar
  • 9,709
28 votes
12 answers

Should we teach functions as sets of ordered pairs?

The context of this question is an "introduction to proofs and mathematics" class for freshman/sophomore math majors. Most textbooks for such a class say something about functions between arbitrary ...
Mike Shulman's user avatar
  • 6,580
14 votes
4 answers

Simpler explanation for finding the vertex of a parabola

I'm tutoring a Grade 11 Math student in BC, Canada, and we're going over parabolas. He's having difficulty with finding the vertex of a parabola - not how to find it, but WHY it works. And I'm having ...
Brandon's user avatar
  • 143
12 votes
2 answers

Why don't we teach codomains of functions in high school?

When I was a university student, I learnt that a function is the data of three informations: the rule that tells how to associate an object $x$ to its image $f(x)$, A domain $E$ where live the ...
Taladris's user avatar
  • 1,491
12 votes
7 answers

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

Background The students in my country are supposed to be able to work with and answer questions about functions at the age of around 15. This is asserted in the standard mathematics curriculum for ...
Improve's user avatar
  • 1,881
8 votes
4 answers

Shifting function graphs: writing vertical offset on the y-side?

Students tend to mix up signs when shifting function graphs around: consider $y=x^2$. To shift it one unit upwards ("increasing $y$"), you write $y=x^2+1$, to shift it to the right ("increasing $x$"), ...
Jasper's user avatar
  • 2,699
7 votes
6 answers

Simple, elegant ways to teach the idea of what functions are for the first time

The context In my country, when the concept of function needs to be introduced in math classes, most teachers will simply talk about $f(x)=c$, $f(x)=ax+b$ and $f(x)=1/x$ (constant, linear and inverse ...
orion2112's user avatar
  • 1,007
6 votes
3 answers

How can you elicit the $\log x = {\log} \cdot x$ error?

You know the error, when you're watching a student work through an algebraic calculation to solve for a variable trapped in the argument of a function, usually $\log$ or a trig function, and you watch ...
Mike Pierce's user avatar
  • 4,845
6 votes
4 answers

Functions, Domains, and Ranges in Precalculus

Possibly related, though of a different flavour. Background In most of the precalculus texts with which I am familiar, readers/students are given a crash course in set theory, handed the definition ...
Xander Henderson's user avatar
  • 8,243
6 votes
4 answers

Is simplifying a rational function considered as a continuous extension?

Given the rational function $f(x)=\frac{x^2-1}{x+1}$. The expression can be simplified to $g(x)=x-1$ and thus the singularity at $x=-1$ is removed. I would personally claim that $f$ and $g$ are the ...
Philipp Imhof's user avatar
4 votes
1 answer

Dissemination of mistakes in international texts such as the IGCSE

I started tutoring IGCSE Mathematics. In preparation for just a 1 hour class for 1 student, I noticed at least 3 mistakes (confirmed with their replies to me) in Chapter 21 of the Haese 0607 book (...
BCLC's user avatar
  • 574
2 votes
3 answers

How can I explain horizontal shifts to a 12-year-old by analogizing with $\text{your money} = \text{my money} + 1?$

My 12-year-old cousin thinks this explanation is the most comprehensible, but she still can't relate the analogy with wealth inequality If I say ...
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