Questions tagged [functions]
For questions about the teaching of the function concept, function properties, and various types of functions.
53
questions
6
votes
5answers
721 views
Does the way we often introduce the concept of a function make sense?
Here are some ideas and a few questions I've been pondering lately related to the teaching of functions in college algebra and precalculus:
Based on my experience, the teaching of functions usually ...
1
vote
5answers
472 views
Proportional density function question
Here is a question I gave on an exam last year. Please let me know what you think of the question (eg: if it is a fair question, easy, difficult, etc). A lot of students were upset for questions like ...
7
votes
2answers
298 views
When working with 12-16 year olds, how should I graph functions when the domain technically isn't $\mathbb{R}$?
Let us assume that I want to graph any of the functions below.
A) A can of soda costs $\$1$. Draw a graph depicting the total cost as a function of the number of cans you buy.
Comment: One cannot ...
12
votes
7answers
8k views
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 ...
17
votes
11answers
2k views
Natural occurrences of a to the (b to the c)?
Are there some natural contexts in which a double exponential occurs,
$x$ to the ($y$ to the $z$):
$$
x ^ {(y ^ z)} \;, \textrm{or} \;, a ^ {(b ^ c)} \; \textrm{?}
$$
Of course one can contrive many ...
2
votes
1answer
193 views
Definition of equation vs. expression vs. polynomial [closed]
I was trying to figure out the distinction of a root and a zero and found people in such discussions make distinctions between equation vs. expression vs. polynomial without defining them. What is ...
20
votes
6answers
4k views
Why is the concept of injective functions difficult for my students?
I was aware that students find the definition of function too abstract and thus find it difficult. However, I thought, once you understand functions, the concept of injective and surjective functions ...
2
votes
2answers
192 views
Would this be a good way of teaching about symmetry?
Symmetry is an important concept in mathematics and it has a built-in aesthetic appeal. The following shows how different types of symmetry relate to one another.
Start with the equation $f(x) = -x^2 ...
1
vote
4answers
358 views
Why is a translated exponential function considered an exponential function?
I am tutoring a student preparing to take Calculus 1 at a university. This student hasn't taken precalculus for a year, so I have been drilling him on definitions, rules, and theorems from a college ...
4
votes
3answers
303 views
Students writing $f(x^2+1)$ when they probably mean $f(x)=x^2+1$
Over the past years teaching freshmen calculus I've repeatedly seen students make the following type of error:
Suppose they have to express some quantity $y$ as function of $x$, when the relation ...
11
votes
2answers
179 views
Confusing verbal descriptions of function transformations
While teaching Function Transformations, I found the verbal descriptions of stretch and squeeze really confusing.
So for $y = f(x)$,
$y = 2f(x)$ is said to stretch $f(x)$ vertically by a factor of $...
3
votes
3answers
743 views
The term "unique" for functions and operations
This is long so...
TLDR: Proposing the math community steer away from the misleading term unique, with respect to functions and algebraic operations. Instead, use unambiguous. Why not? Analysis below....
45
votes
21answers
6k views
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 ...
9
votes
3answers
536 views
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. ...
3
votes
2answers
116 views
Quadratic modeling project with upward-facing parabola
I'm teaching a college algebra course and I'm trying to design a few projects that involve modeling with quadratic functions. So far I have two ideas that involve downward-facing parabolas (projectile ...
2
votes
3answers
312 views
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 ...
12
votes
3answers
553 views
Examples (for beginners) of real functions which are not given by elementary formulae
Question: What are good examples of functions $f: \Bbb R \rightarrow \Bbb R$ (or $f: D \rightarrow \Bbb R$ with $D \subseteq \Bbb R$) which are not just given by "a formula" (or finitely many formulae ...
1
vote
1answer
204 views
How can I make "complex" graphs that combine multiple functions with a software?
Til today I've been using geogebra to sketch functions for my students quizzes or homework. Sometimes I use the ones that I found searching in google, but this takes a lot of time specially because I ...
7
votes
6answers
549 views
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 ...
8
votes
4answers
933 views
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 $x$"), ...
17
votes
8answers
8k views
"Real world" examples of implicit functions
When teaching implicit differentiation in freshman calculus I lack good examples which might help students relate the theory to applications in other sciences.
So I'm looking for (relatively simple) ...
-1
votes
3answers
186 views
Which book has functions and their respective graphs? [closed]
I am looking for a book, which has different many different types of functions and their graphs (like, Weierstrass function, Takagi function).
3
votes
1answer
125 views
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 (...
13
votes
4answers
309 views
How can we focus students on the various data types in multivariable calculus?
To try to find out if students knew what the gradient was, after the computational questions, I asked the following question on an exam:
Let $f(x, y) = 5 - x - y$. Why doesn't it make sense to find $\...
28
votes
10answers
10k views
Why do we teach even and odd functions?
I've been either a student or an instructor in Precalculus or Calculus 1 at about 6 institutions now, and teaching the definition of even functions (where $f(-x) = f(x)$) and odd functions (where $f(-...
6
votes
4answers
822 views
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 ...
8
votes
1answer
260 views
What is the name of this discipline in mathematics education?
I am struggling with my students who can think only in concrete terms,
they can compute with concrete numbers but are not able to think in terms of e.g. functions on natural numbers and come up with ...
15
votes
7answers
817 views
How should I introduce the concept of a function to a precalculus student?
My brother has not taken a math class in $10-15$ years. He is studying for the GRE so I have been teaching him a chapter or two from my precalculus book. So far, he has learned (and excelled at) basic ...
1
vote
1answer
4k views
Project Based Learning (PBL) ideas for secondary math topics
I have searched the web but not found too much that fits what I want. So, I thought I would try asking here.
I am looking for projects to use as PBL for grade 10 math. Specifically for the topics of:
...
6
votes
3answers
579 views
Solid of revolution simulator?
I'm searching for an offline portable program that demonstrates the concept of how does a function revolve around X or Y axis It should be similar to this: Solid of revolution simulator
but I want to ...
2
votes
2answers
249 views
How to explain codomain of a function?
A possible explanation is that some yet unsolved advanced functions have a known codomain, but not a known range.
Got any better ideas?
18
votes
4answers
553 views
"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 $f$ ...
9
votes
2answers
314 views
How to explain the range to students?
I mostly tutor students ranging from beginning algebra to calculus level. I think the explaining the range as "the set of all possible outputs" would not really cut it for someone struggling with math ...
7
votes
3answers
265 views
Explaining the domain of a function to students?
I mostly tutor community college students ranging from beginning algebra to calculus level. There are several ways in which I explain the domain:
"The domain of a function is all the values you can ...
15
votes
5answers
2k views
For purposes of teaching, should constant functions be considered "linear functions"?
I can see arguments both for and against classifying constant functions as linear functions.
Against: "Linear function" means "first-degree polynomial function", and constant functions are not first-...
24
votes
5answers
2k views
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) = \...
3
votes
3answers
200 views
Which math class should I take as an exchange student in the USA (OH)? [closed]
I will go to an American High School (Ohio) this summer for a year and I will probably be a junior. I got a list of all classes, but there are really many classes to choose especially math classes. I ...
4
votes
1answer
131 views
In Polynomial Form, After Simplification (But Not Before!)
We know that $\dfrac{(x^2+1)x^3}{x^2+1}=x^3$ and $(\sqrt{|x|})^4=x^2$ for every $x\in \mathbb{R}$. Can $\dfrac{(x^2+1)x^3}{x^2+1}$ and $(\sqrt{|x|})^4$ be called polynomials?
Is there a general name ...
6
votes
2answers
269 views
Name to use for codomain/range/target
There are many questions about how best to teach functions, for example why don't we teach codomains in HS and should we teach them at all. On Math.SX there are questions about the "right" name for ...
6
votes
4answers
365 views
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 ...
22
votes
2answers
2k views
Is this just a mistake or a more serious misconception?
One of my main research areas is algebraic thinking at different levels. Yet, from time to time, I observe something that I cannot relate to anything else that I know.
This is the story of one of ...
13
votes
4answers
933 views
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 ...
3
votes
1answer
382 views
What are sources for non-routine problems involving quadratic functions (in one variable)?
I'm planning to get some sources which explain beautiful problems about quadratic function. I know that there are another kind of functions, but the quadratic function has different applications in ...
27
votes
12answers
3k views
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 ...
14
votes
1answer
165 views
Resources suitable for a beginners' course with exponentials
I'm currently involved in developing materials for a new UK tier of examination known as Core Maths. The course is designed for 16-18 year olds to further their mathematics education but without ...
12
votes
2answers
478 views
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 ...
10
votes
2answers
274 views
What is a good way to explain the slightly different kinds of continuity?
What is a good way to explain the slightly different kinds of continuity to students? I have in mind these kinds of continuity:
A function is continuous at a point. (This also has two sub-kinds: ...
8
votes
3answers
377 views
Examples of when $\tan(x) = \frac{\sin(x)}{\cos(x)}$ is useful
I'm making a video right now about the unit circle definitions of the basic trig functions.
I've done sine and cosine, and am now talking about the tangent function.
As most of you know, it can be ...
12
votes
5answers
9k views
Real life examples to motivate the study of linear functions
Some years ago I used mobile phone or internet rates (for example, with basic fees and a given charge per minute or by data volume) to introduce and motivate the study of linear functions.
However, ...
9
votes
2answers
1k views
Why are hyperbolic functions given "short shrift" at "low" levels of math?
Starting with "precalculus," students learn trigonometric functions. After they've spent a semester or more learning how to differentiate and integrate these trigonometric functions in calculus, ...
