"the function f where f(x)=3x+4"

[Sarah Stevens]

We seek clarification on a paragraph on page 7 of the progression. We were surprised to see this distinction made between f and f(x). We absolutely agree that f(2)=0 is referring to a single output 0 when the input value is 2. We are struggling with understanding why  f(x) isn’t referring to the entire function. For every x value I input, I would get 1 output. If I plotted all the possible inputs and outputs on a graph I would get the graph of the function f(x). It seems as if I should be able to call it f(x).

To further confuse matters the graph adjacent to this statement uses the labels f(x) and g(x). Is it ok to use that label for a graph but every where else we should only use f and g?

Another problem we have is when we seek additional information from Illustrative Math on these notation issues. Within the Functions tasks, sometimes they use functions notation and sometimes they use “y=” (for example http://www.illustrativemathematics.org/illustrations/635). The top of pg 3 in the Algebra progressions suggests we are the right track to seek clarification on the notation for functions vs the notation for equations and if it is allowable for students to fluidly move between the two. I didn’t find any further insight in the functions progressions on switching between notations, when its appropriate, and when it isn’t.

Thanks again for your help as we work on correctly processing this information!

[Bill McCallum]

First, thanks for pointing out the error in the labeling of the graphs, it should say either $f$ or $y=f(x)$. The fact that the error crept in is an example of just how widespread this usage is, and maybe it is futile to fight it. But it can’t literally be true that $f(x)$ is a function, because it’s a number, and a number is not a function. The letter $x$ refers to a specific but unspecified number in the domain of $f$, and $f(x)$ refers to the corresponding output. That’s the way function notation works. I would worry that not being precise in this usage leads to confusion and misconceptions later on. I think your desire to use $f(x)$ to refer to the function comes from a sense that $x$ in some way represents all the input values at once. But this itself is dangerous, I think: a lot of the trouble students have with algebra comes from a feeling that $x$ (or whatever letter you are using) isn’t really a number but is some vague mystical thing they have to perform mysterious rites on. So the more we can keep students anchored in the idea that the letters in algebraic expressions and equations are just numbers, and that the things you do to expressions and equations are just the things you can do to numerical expressions, the better.

As for the $y=$ notation, when we say something like “the function $y=x^2$” we are using abbreviated language for “the function defined by the equation $y=x^2$, where $x$ is the independent variable and $y$ is the dependent variable.” You can’t say that every time, so we have a shortened form, which depends on certain conventions: the dependent variable occurs on the left and an expression in the independent variable occurs on the right. So, it would be problematical to say “the function $2p+3q=5$” because it doesn’t specify which variable is the independent variable and which the dependent variable, and because, although this particular equation is solvable for one or the other, that’s not always true with an equation in two variables.

With these conventions, the terminology “the function $y=f(x)$” is o.k. If $f$ is the function, then $f(x)$ is the expression in $x$ giving the value of the function at $x$.

[Alexander]

A welcomed clarification to function notation. This idea stumped many of my classmates during both my undergraduate and graduate courses. Much better than having to resort to f(x_1) to make clear the reference to a single output. Don’t get me wrong, students should still be comfortable with x_1, x_2, ….x_n inputs in the correct context and while reading textbooks.

With this change in mind, would the example task change?

Page 8, Cell Phones

Let f be the number of people….

 

Notation should be used to improve communication to all students and not be an esoteric rite of passage. I’m not that worried about job security.

[Bill McCallum]

I’m not sure what the suggestion is here: the task says “let $f(t)$ be the number of people, in millions, who own cell phones $t$ years after 1990.” This strikes me as correct: $t$ is a number (an input to the function $f$) and $f(t)$ is a number (the corresponding output). You could also say something like “let $f$ be the function such that $f(t)$ is the number of people ….” But it seems an acceptable abbreviation to say it the way it is in the task. It wouldn’t be correct to say “let $f$ be the number of people …” because $f$ is not a number, it’s a function.

[Alexander]

Not a suggestion, mostly just personal clarification using an example in the book before I meet with teachers. You cleared it up, thank you. I didn’t pay attention to the phrase “be the number” as an obvious reference to the output value.

Example f(m)=2m+4

f is a function that related the amount of money paid to a taxi based on the number of miles traveled.

f(m) is the amount of money paid to a taxi after traveling m miles.

[Bill McCallum]

Thanks for the clarification. Your first example illustrates the difficulty in describing a function abstractly; you might also say “f gives the amount of money paid to a taxi as a function of the number of miles traveled.” Also, I would add units and say “the amount of money (in dollars).”

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