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One of the major themes of precalculus is what I call “connecting geometry to algebra”. Being able to translate between an algebraic statement like $f(x)= f(-x)$, and the geometric statement that the graph of $f$ is symmetric about the vertical axis is a great instance of this. This is just one more way to practice reinforcing function concepts, and the ...
37
In the U.S. Common Core standards, functions are supposed to be introduced in the 8th grade, i.e., around age 13-14. So arguably age 15 is a year or two behind where they ought to be.
The standard for the 8th grade says:
Understand that a function is a rule that assigns to each input
exactly one output.
So honestly that really doesn't seem like a hugely ...
32
It's not really a question about functions and domains, but about valid expressions. The question is, "For which real numbers $x$ is the following expression valid/well-defined/well-formed?" So, for example, $\frac{x + 1}{x^2 - 1}$ is valid exactly for $x \notin \{-1, 1\}$, and $\sqrt{x}$ (assuming this notation is established to mean the real-valued square ...
31
I think you will find that almost everyone has this problem when they first starting to learn rigorous mathematics, and many students will never overcome this difficulty.
The following three statements are logically equivalent to each other, but students will almost all find the first to be easier to understand than the second, and will not be mature enough ...
27
If your students have learned some statistics, then you could point out that the normal distribution's probability density function uses this double exponential.
$$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right)$$
24
You might remind them that $y$ is just a name for a number. When they draw a plot, they draw a bunch of points: maybe $y=3$ here, $y=5$ there, and $y=-2$ over there. But at some point (no pun intended) we want to talk about the entire shape: we want to say that $f$ is symmetric, that $f$ is concave, that $f$ has an asymptote. We can't do that with $y$; ...
23
Start by talking about functions in general, not only about functions that can be expressed by a simple formula in x and y. Examples:
The function that maps every non-empty list to its first element.
The function that maps every finite set to its size.
The function that maps color names to RGB triples.
The function that maps days to sunrise times at a ...
22
I wanted to provide a different perspective on this. I just recently graduated with an undergraduate degree in mathematics, and I wanted to say a few things about what personally helped me in understanding functions, and why I think introducing them as sets of ordered pairs is important.
Before I took my first proofs class, I thought of functions the ...
21
You should tell them these two main benefits:
(1) Function notation is concise! For example, instead of writing "Find $y$ when $x=5$" one can simply write "Find $f(5)$" This becomes very appreciable when dealing with long or complicated problems asking for a lot of information. We also shorten things like this all the time. For instance, instead of writing $...
21
In real-world applications, the typical case is that the domain is neither implicit in an expression we write down, nor explicitly stated along with the expression. Rather, one uses knowledge of the real world to decide what numbers make sense as inputs to the function. So for example, if $x$ represents an investment, and $f(x)=x-3$ represents the profit on ...
20
On the other hand, they are really struggling with injective functions. Even after spending a lot of time, they often say "a function is one-one if every element in the domain has a unique image".
Have you asked your students what they mean by "unique" when they say that?
The reason I'm asking is that, by some commonly used definitions ...
19
In my experience, the "set of ordered pairs" is a difficult and confusing definition. Moreover, I feel it's almost totally unnecessary because in practice in mathematics we do always treat functions as rules. No one ever defines a function as $f = \{ (x,y) \mid y = x^2 \}$; we always write $f(x)=x^2$. Of course, "rules" in this sense are much more general ...
18
Here you can see that knowing if the function is even or odd can help you when you are integrating over the interval $[-a, a]$.
You can reduce really-hard-to-look-at integrals to zero just by knowing this. As an example, to calculate $E(Z)$ where $Z \sim N(0, 1)$, the standard normal distribution, you have:
$\displaystyle E(z) = \frac{1}{\sqrt{2\pi}}\int_{-...
17
Let me give you yet another point of view which is a bit closer to discrete mathematics, that is, from introduction to functional programming. As this course is very function-heavy, it was worth to spend the very first class solely on the basics of functions. Please note that what I describe below builds quite different intuition than that of continuous ...
17
It seems clear that there is a certain conceptual gap in the student's understanding. My suspicion is that the student is essentially running the following program in his mind:
Initialize factorial = 1
For $j = k$ to $1$
factorial = factorial * (2j - 1)
Output factorial
(The correct program should read: for $j = 2k - 1$ to $1$, factorial = factorial *...
17
user52817, in the comments, has this exactly right: In the context of Precalculus, the implied domain of a function is the largest subset of $\mathbb{R}$ on which the function is defined. So introduce that notion, and practice it with questions like "What is the implied domain of $f(x)=\sqrt{x}$?" (You can also let your students know that sometimes people ...
16
Besides applicability in topics like integration and Fourier analysis, it also connects algebra to calculus at least in the way that multiplication of even/odd functions behaves like addition even/odd numbers:
Multiplying two even functions gives an even function.
Multiplying two odd functions gives an even function, too.
Multiplying an even
and an odd ...
16
The crucial thing the students need to realise is that the (e.g.) $x$ that turns up in the function definition is a bound variable. That's what allows it to be freely renamed or indeed omitted without changing the semantics.
Unfortunately, education tends to completely obscure this facet by a) always using the same dumb variable names as if there were a ...
16
Functions are far broader and more applicable than you give them credit for. Consider the following:
Country or state
Capital
Elevation (in meters)
Bolivia
Sucre
2783
Ecuador
Quito
2763
Colombia
Bogata
2619
Eritrea
Asmara
2363
Ethiopia
Addis Ababa
2362
Mexico
Ciudad de Mexico
2216
New Mexico
Santa Fe
2152
Wyoming
Cheyenne
1856
Colorado
Denver
1613
...
15
A linear function is not necessarily a first degree polynomial function: zero function is also linear.
In France the terminology is more appropriate than the traditional English one: a linear function is a function of the form $f(x) = ax$, while a function of the form $f(x) = ax + b$ is called an affine function.
So, strictly speaking, constant functions ...
14
Learning to think about functions abstractly should be one goal in precalculus, and function symmetry helps. Also suppose we carefully protected a student from knowing anything about function symmetry. Upon learning about flux in vector calculus, would this student be able to quickly see that the flux of the vector field ${\bf F}(x,y,z)= y^2{\bf j}$ through ...
14
Because x and y are just variable names
It happens that sometimes y=f(x), but other times z=f(x,y), w=f(x,y,z), or x=f(y) for that matter. All of these variable names are syntactically equivalent, and the mere existence of "x" and "y" in an equation does not necessarily connote that "x" is the independent variable and "y" the dependent. Thinking of the ...
13
As per my comment above, it looks to me like the student saw $2k−1$ which is a pretty standard mathematical way to say "odd numbers" and then saw the factorial and thought to themselves "looks like i need to factorial only the odd numbers". While I have never encountered this particular misconception, I categorize it into a general category of an "illogical ...
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 ...
13
We give high-schoolers many different explanations of the word "function." Here are a few that are either implied or outright stated at various points in a student's education:
A function is an expression in terms of $x$. This is pretty unusual all by itself, but it may appear in conjunction with (2) or (3) (i.e. students may reason that $f(x)$ or ...
12
While I commented otherwise, let me still give an answer which complements others, relates to a more general question, and answers some starting discussion in comments.
We often consider, as mathematician, that defining everything very precisely is important, and it truly is. But what is often done, and in my opinion is often a mistake, is to consider that ...
12
Although there are good answers already, I think the ambiguity of the vocabulary is being underestimated. This might be because French definitions have more ambiguity than others, I can't tell.
In France, at least up to some years ago, the words function ("fonction") had various meaning depending on the context.
In high school (and probably some colleges)...
12
While I haven't done a systematic survey, my impression is that the overwhelming majority of pre-calculus and calculus texts define a linear function to be one of the form $f(x) = mx + b$ with no stipulation that $m \neq 0$. Thus, if you define linear to be a polynomial of degree 1 you are likely to be contradicting whatever textbook you are using.
From a ...
12
How about $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = y$ where $y$ is the unique solution to $y^5+x^2y+5=0$? This does not have an elementary formula, but students can understand that for any value of $x$ the function $g(y) = y^5+x^2y+5$ is increasing (and the limits at $\pm \infty$ are $\pm \infty$), so it must have a unique root for each $x$. You ...
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