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 ...
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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(-...
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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 ...
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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) = \...
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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 ...
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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 ...
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"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 $f$ ...
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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 ...
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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 ...
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"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) ...
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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 $\...
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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-...
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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 ...
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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, ...
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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 ...
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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 ...
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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 $C(f,g) = f∘g$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 $...
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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: ...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 $x$"), ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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:
...
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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 ...
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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, ...
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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 ...
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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 ...
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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 (...
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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....
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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 ...
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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 ...
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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 ...
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