Questions tagged [functions]

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

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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
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31 votes
10 answers
11k 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(-...
Nick Matteo's user avatar
28 votes
12 answers
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 ...
Mike Shulman's user avatar
  • 6,540
24 votes
6 answers
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) = \...
Paul Castle's user avatar
24 votes
5 answers
814 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 $f$ ...
Michael Bächtold's user avatar
21 votes
6 answers
5k 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 ...
Divakaran Divakaran's user avatar
21 votes
2 answers
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 ...
Amir Asghari's user avatar
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18 votes
5 answers
4k views

Why is there a disconnect in the usage of "domain" between high school and higher mathematics, and where does it come from?

In high school (in the US, at least), it is common to define the domain of a function as the set of real numbers for which the function is well-defined and returns a real result. Then students are ...
Reed Oei's user avatar
  • 412
17 votes
11 answers
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 ...
Joseph O'Rourke's user avatar
17 votes
8 answers
13k 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) ...
Michael Bächtold's user avatar
17 votes
5 answers
436 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 $\...
Chris Cunningham's user avatar
15 votes
4 answers
2k views

Why do we use functional composition in the order we do?

Function composition means, roughly, taking the output of a function and applying it to the input of another function. If we define an object C to represent this operation, we could say $C(f,g) = f∘g$ ...
David Lalo's user avatar
14 votes
5 answers
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-...
mweiss's user avatar
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14 votes
7 answers
991 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 ...
Ovi's user avatar
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14 votes
4 answers
2k 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, ...
Tom Au's user avatar
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14 votes
4 answers
2k 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 ...
Brandon's user avatar
  • 143
14 votes
1 answer
188 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 ...
Karl's user avatar
  • 1,552
13 votes
1 answer
222 views

Teaching functions/mappings early

Functions and mappings are usually introduced late in the curriculum, and functions of arity two or more are considered "advanced" (many don't even see them before college). On the other hand, the ...
dtldarek's user avatar
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12 votes
7 answers
9k 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 ...
Improve's user avatar
  • 1,881
12 votes
14 answers
6k views

Examples of relations that are not functions

When teaching functions, one key aspect of the definition of a function is the fact that each input is assigned exactly one output. I always felt that the "exactly one" part is confusing to ...
Jasper's user avatar
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12 votes
3 answers
730 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 ...
Torsten Schoeneberg's user avatar
12 votes
5 answers
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, ...
Julia's user avatar
  • 1,245
12 votes
2 answers
576 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 ...
Taladris's user avatar
  • 1,367
11 votes
2 answers
308 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 $...
reflectionalist's user avatar
10 votes
2 answers
311 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: ...
Rory Daulton's user avatar
  • 2,582
9 votes
2 answers
366 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 ...
Ovi's user avatar
  • 725
8 votes
3 answers
428 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 ...
Alec's user avatar
  • 828
8 votes
5 answers
992 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 ...
Brain Gainz's user avatar
8 votes
1 answer
283 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 ...
Gergely's user avatar
  • 183
8 votes
3 answers
1k 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. ...
Eleanor Hankins's user avatar
8 votes
4 answers
2k 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 $x$"), ...
Jasper's user avatar
  • 2,689
7 votes
7 answers
1k views

Write $y=\sqrt{3+x}$ as the composite of two functions

For the question "Write $y=\sqrt{3+x}$ as the composite of two functions", what if a student gives the answer $f(x)=\sqrt{3+x}$ and $g(x)=x$? This answer would be technically correct but it ...
Zuriel's user avatar
  • 4,265
7 votes
2 answers
483 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 ...
Improve's user avatar
  • 1,881
7 votes
6 answers
836 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 ...
orion2112's user avatar
  • 1,007
7 votes
3 answers
362 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 ...
Ovi's user avatar
  • 725
6 votes
3 answers
845 views

Composite functions

How would you describe the existence of a composite function $f(g(x))$in terms of range of $g$ and domain of $f$ . Does range of $g$ need to be subset of domain of $f$ or is it sufficient if the two ...
Janaka Rodrigo's user avatar
6 votes
3 answers
467 views

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
  • 4,777
6 votes
4 answers
454 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 ...
Philipp Imhof's user avatar
6 votes
4 answers
1k 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 ...
Xander Henderson's user avatar
  • 7,697
6 votes
2 answers
370 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 ...
kcrisman's user avatar
  • 5,980
6 votes
2 answers
722 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: ...
J W's user avatar
  • 4,635
6 votes
3 answers
652 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 ...
Seraj's user avatar
  • 61
5 votes
3 answers
3k views

What are some examples of great functions that are not too elementary (easy)?

I am teaching precalculus/basic calculus to my class (high schoolers of around 18 years of age), and I'm always searching for nice functions to plot and study (finding the domain, the function's sign, ...
marco trevi's user avatar
5 votes
4 answers
347 views

Educational resources commonly address slant asymptotes. Why not general polynomial asymptotes?

Back in 2018, I wrote a post about asymptotes of rational functions in which I addressed not only horizontal and slant/oblique asymptotes, but also the general case of "polynomial asymptotes.&...
Justin Skycak's user avatar
5 votes
3 answers
184 views

Sources of sample data for regressions

I'm looking for sample data to give algebra 2 students to teach about using Desmos to do regressions. Some example data sets folks at my school already have compiled are: Number of Lego pieces vs ...
DreiCleaner's user avatar
4 votes
5 answers
782 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 ...
Eleven-Eleven's user avatar
4 votes
1 answer
142 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 ...
Behzad's user avatar
  • 2,363
4 votes
3 answers
438 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 ...
Michael Bächtold's user avatar
4 votes
1 answer
200 views

Why is there variation in the meaning of "Standard form" for a quadratic?

I'm teaching this year out of "Precalculus with limits" by Ron Larson [7th ed], and the following expression appears in the unit introducing polynomial functions: $f(x)=a{(x-h)}^2+k$ He ...
Cassius12's user avatar
  • 419
4 votes
1 answer
145 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 (...
BCLC's user avatar
  • 574