Questions tagged [functions]

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

Filter by
Sorted by
Tagged with
4
votes
2answers
81 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 ...
3
votes
5answers
2k 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
186 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
188 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
268 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
289 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
163 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
675 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....
44
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
487 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
108 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
310 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
538 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
174 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
502 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
816 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
7k 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
181 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
124 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 (...
12
votes
4answers
303 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 $\...
29
votes
10answers
9k 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
756 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
254 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
784 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
3k 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
555 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
240 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
536 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
282 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
259 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
197 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
128 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
237 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
354 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
688 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
369 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 ...
26
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
162 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
442 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
260 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
372 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, ...
2
votes
1answer
300 views

Is there a conventional function notation that takes a polynomial and order and yields the coefficient corresponding to the order?

I am writing a book and for the sake of simplicity I want to do something as follows. Coef((-3x^2 +5x -1)(x^2 +1), 2) = -3 -1 = -4 where the first argument ...
6
votes
2answers
632 views

Preimage of a set under a function

I am looking for suggestions about ways to introduce the preimage of a set under a function. My experience is that many students find it a confusing concept. The definition I use is as follows: ...