13

I don't think there's a lot of educational value to fixating on trying to word definitions in exactly the perfect way. Students have trouble with the notion of a function because it's hard. The way they're going to get a handle on it is by struggling with it, encountering the hard parts of the definition, and finding and eliminating their misconceptions ...


6

To start with an opinion, I think that this classification exercise is kind of silly. The student is being asked to put functions into some categories without having a clear idea about what those categories mean or are used for. We introduce definitions and categorizations in order to help us understand abstract ideas. A definition without the underlying ...


5

I personally use the following terminology: A relation $R \subset A \times B$ is said to be single-valued if $(a,b_1) \in R$ and $(a,b_2) \in R$ implies $b_1 = b_2$. A relation $R \subset A \times B$ is said to be total if for all $a \in A$ there exists $b$ in $B$ with $(a,b) \in R$. A relation which is both single valued and total is a function.


5

In Germany, we already do this. A function is introduced as an unambiguous mapping in 7th grade (~13 years). While I don't have any data on this, I doubt that German students do significantly better due to this choice of words.


3

The working definition I have in my head doesn't fit the more rigorous definitions others have put in their answers. I think of exponential growth and decay as being constant percentage growth or decay from or toward an asymptote. My favorite example is temperature of an object, which is shifted with the ambient temperature being the asymptote. I use y = a*b^...


3

I say the key descriptor of a exponential function is constant multiplicative rate of change, much as the descriptor of a linear function is constant additive rate of change. The function $f(x)=a(1.5)^x$ increases by 50% when $x$ increases by 1: $$\frac{f(x+1)}{f(x)} = \frac{a(1.5)^{x+1}}{a(1.5)^x} = 1.5$$ But adding a non-zero constant changes that: $$\frac{...


3

For me, the standout problem here is this: relations are verbs. Without a relation/verb, you don't have a statement, you have the equivalent of a sentence fragment. What the student has written in this case isn't even wrong, it's just malformed nonsense. I try mightily to get students to at least see, at first pass, that a piece of writing without any verb/...


3

Many answers already, so I'll keep this one short: it has been realized by researchers in didactic that one difficulty in the concept of function is that it changes status: at first each function is considered as a process (a verb in @ΦDev's answer); they meet several of them, each being akin to a (unitary) operation, not very different from addition or ...


2

To answer the question of why you need f... ... have them consider the region between two graphs. Unless you have a way to distinguish between the different y values, you're going to be hopelessly lost. Now you don't need to use f... you could use subscripts: $y_{1}, y_{2}$ (and in fact, that's how graphing calculators handle it). But it's nice to ...


2

Well... $ \def\zz{\mathbb{Z}} $ Let $f : \zz → \zz$ defined by $f(n) = n+1$ for every $n∈\zz$. Then $f(0) = 1$ and $f(f(0)) = 2$ and $f(f(f(0))) = 3$ and so on. It is now obvious why having functions as first-class objects is useful, since we can repeatedly apply them. Similarly, the Mandelbrot fractal is defined in terms of iterating an elegant function.


2

I've found terminology differs as well, but this is how I think of the phrase "by a factor of" for $af(x)$ If $0<|a|<1$ then it is a vertical "squeeze" to get a factor of $a$ between the new and old $y$ values If $|a|>1$ then it is a vertical "stretch" to get a factor of $a$... And then a point $(x,y)$ becomes $(x,ay)$. So $a$ is the new scale or ...


1

Suggest teaching students about composition of functions. Show them that if $f(x)=x^2+1$, then $f(2x)= (2x)^2 +1$ and $f(x^2 + 1)$ = $(x^2+1)^2+1$ If they do enough problems with composition of functions, they will be less likely to make this mistake again. This should be more effective than explanation about what the notation means.


1

If I continually saw this mistake then I'd try to get the idea of $y=f(x)=x^2+1$ more solidified: "$y$ is a function of $x$ and that function can be represented by an equation." I'd probably try to incorporate functions and transformations into the first week of homework so that students know the difference between $f(2x)$ and $2f(x)$ and $f(x)=2x$. One ...


1

I tend to be very strict on how I define things in this area - I like to keep things formal in preparation for University study. I'll explain everything I use and why I believe it to be effective - and use your examples. Let $ y = f(x) $. $ y = 2f(x) $ is said to be a vertical stretch of scale factor $2$ parallel to the y-axis. $ y = \frac{1}{2} f(x) $ ...


1

You could start with two tables of values for x (the input variable) and y (the output value) in both. To start, each should represent a permutation on say, the set {1, 2, 3, 4, 5}, but don't use the word "permutation." Label one "Table A", the other "Table B". For each line in Table A, introduce the notation A(1)=, A(2)=, etc. Similarly, for Table B. ...


1

Defining a function as a set of pairs is a lot more accessible than it might seem. Think of a directory in a building, which lists people's names and the room number of each person's office. (Assume everyone's name is different, but people might share an office.) The directory spells out an assignment, $d$, of names to numbers. Name $x$ is presented next ...


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