Questions tagged [functions]
For questions about the teaching of the function concept, function properties, and various types of 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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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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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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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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"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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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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Manipulative materials to teach functions
I am looking for manipulative materials to teach functions (the concepts including domain, image, etc.) and kind of function (affine, quadratic, exponential logarithmic, polynomial, trigonometric) ...
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The two paradigms of seeing a functions
When we are first taught functions , we are typically taught of them as maps between real numbers and we taught to think of them mainly as a mapping between elements. It seems intuitive to take this ...
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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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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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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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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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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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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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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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What is the most fundamental continuous function in calculus after a constant (totally straight) line?
What would be to teach in most countries and education systems?
(Taught before "high education frames", i.e. before doing bachelor of arts in mathematics).
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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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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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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 ...
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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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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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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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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 ...
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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:
...
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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 ...
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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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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 ...
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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 ...
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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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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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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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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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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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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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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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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 ...
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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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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 ...
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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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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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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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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).
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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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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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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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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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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?
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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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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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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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