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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-degree.
  • For: "Linear function" means "polynomial function of degree at most 1, and constant functions do match that definition.
  • Against: But we don't define "quadratic function" to mean "polynomial function of degree at most 2"; if we did, then linear functions (under whatever definition) would also be quadratic, and nobody uses the word "quadratic" that way.
  • For: Nevertheless, the graph of a constant function is a line, so it's a linear function.
  • Against: But the graph of $x=c$ is also a line, and that's not a function at all, let alone a linear one!
  • For: The notion of "slope" is well-defined for constant functions, and the slope just happens to be $0$.
  • Against: See above re: quadratic functions. Writing a constant function in the form $y=mx + b$ with $m=0$ is analogous to writing a linear function in the form $y=ax^2 + bx + c$ with $a=0$. Yes, you can do that, but we generally don't.

To some extent I can see this as just a matter of taste or convention. Does anybody see any compelling arguments one way or another that I have overlooked? Is there a "best" approach?

(Background: I recently put the following True/False question on an exam:

If a function is linear, then it is invertible.

Whether this is true or false depends on whether you regard constant functions as "linear" or not.)

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    $\begingroup$ If I'm not mistaken, some people define linear function as something that can be expressed as $y=mx$. That is, some do not consider $y=mx+b$ with $b\ne 0$ as a linear function. (See Wikipedia, for example.) $\endgroup$
    – JRN
    Jul 25, 2016 at 3:59
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    $\begingroup$ It's worth bearing in mind that in linear algebra $y=mx+b$ is linear in $b=0$, but $m=0$ and $m\neq 0$ are both fine. If students are close to reaching a linear algebra course, they may get confused if you tell them $m\neq 0$ is required. $\endgroup$
    – Jessica B
    Jul 25, 2016 at 5:57
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    $\begingroup$ What did you mean to test with your True/False question? It has been argued in the answers that the question might be too subtle, I would rather say that if you are not sure which definitions feels best, questions whose answers depends on these level of precision in the definition have little relevance. $\endgroup$
    – Benoît Kloeckner
    Jul 25, 2016 at 14:03
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    $\begingroup$ Seems like a potentially interesting discussion to have with students... $\endgroup$
    – Benjamin Dickman
    Jul 25, 2016 at 15:05
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    $\begingroup$ @mweiss: sidetracking a bit, testing whether they understand what invertible means would be more accurately measured by asking "which of the following functions are invertible" and listing half a dozen examples. Here in practice you measured a mixture of understanding invertibility, of knowing the (given) definition of linear functions, and of not forgetting about constant functions if they where part of your definition. It seems much more difficult to interpret. $\endgroup$
    – Benoît Kloeckner
    Jul 25, 2016 at 16:26

5 Answers 5

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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 are affine functions, but in the traditional English terminology they are also linear functions (though they do not preserve linear combinations).

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    $\begingroup$ The French terminology better fits which the term "linear map" in linear algebra. I like it very much... $\endgroup$
    – Stephan Kulla
    Jul 25, 2016 at 9:14
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    $\begingroup$ Same in Spanish: afín $\endgroup$
    – bgusach
    Jul 25, 2016 at 9:47
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    $\begingroup$ At the university level, in the USA at least, the distinction between linear and affine is introduced. This brings us to a central quandary of math education: there is a lot of throwing out of what we learned in the first five to ten years to have it replaced with a refined version. Is that a necessary aspect of math education or are we wasting our time with teaching things "wrongly"? $\endgroup$
    – Todd Wilcox
    Jul 26, 2016 at 3:47
  • $\begingroup$ Don't remember coming across the term affine through linear algebra and PDE in a math minor, so unless it was brushed past\forgotten, or reserved for even later courses, I wouldn't say that your statement is universally true. As to replacement, I certainly agree, and wish to teach it "right" the first time, but often the complexities of things require a somewhat simplified way of looking at things to fit reasonably into high school (I believe some stuff details of quantum shapes... and Newtonian physics are the same... useful, but in their finality, "wrong" when the full details are covered). $\endgroup$
    – JeopardyTempest
    Jul 26, 2016 at 6:05
  • $\begingroup$ "Affine" comes from affine spaces (like a line or a plane in classical geometry, viewed as a collection of points, on which vectors can act). In Russia, functions $f(x) = ax + b$ are also called linear. $\endgroup$
    – Alexey
    Jul 26, 2016 at 8:32
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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 more abstract point of view, if you define linear functions as polynomials of degree 1 then you lose the nice correspondence between linear functions and affine self-maps of $R$. You also lose closure under addition and scalar multiplication, and thus a nice example of a vector space over $R$. A little less abstractly, you also lose the fact that the antiderivative of a constant function is always linear.

For these reasons (as well as the other reasons you enumerated) I would go with defining linear functions in such a way that it allows for constant functions.

Having said all that, I think that your True/False question is a bit too subtle for a lower-level class (unless you had really emphasized the point).

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    $\begingroup$ @SteveJessop I'm in the middle of a 5-minute break in a summer Applied Calculus course. I just checked our book for that course (Brief Calculus by Goldstein et al) as well as Stewart's Calculus book. They both define linear in such a way that constant functions are included but they both define quadratic in a way that explicitly rules out a = 0. $\endgroup$
    – John Coleman
    Jul 25, 2016 at 13:38
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    $\begingroup$ ^ I agree with the preceding, every book I've ever seen defines quadratics as $a \neq 0$. And the Ratti/McWaters Precalculus book I just checked also includes constants as linear equations (explictly allows $m = 0$). $\endgroup$
    – Daniel R. Collins
    Jul 25, 2016 at 14:02
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    $\begingroup$ If "quadratic" (polynomial) requires that the highest-order term be non-zero, then the sum of two such may fail to be such. Similarly for "linear" (polynomials). That would be awful, no matter what textbooks say. $\endgroup$
    – paul garrett
    Jul 25, 2016 at 18:08
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    $\begingroup$ @paulgarrett It seems natural to reserve "quadratic" for expressions to which you can apply the quadratic formula and whose graphs are parabolas. In any event, the terminology is standard. Mathematical language, like natural languages, don't always evolve in a strictly logical way. $\endgroup$
    – John Coleman
    Jul 25, 2016 at 18:16
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    $\begingroup$ @paulgarrett My impression is that using "quadratic" as a synonym of "degree 2" is widespread, and not just in elementary texts. Similarly for "cubic", "quartic" and "quintic". Not that it proves much, but is is of some significance that Wikipedia's entry on the quadratic has the $a \neq 0$ stipulation (although it does mention that "some authors" use the alternate convention): en.wikipedia.org/wiki/Quadratic_function $\endgroup$
    – John Coleman
    Jul 25, 2016 at 19:16
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A "linear function" is a function satisfying both

$f(x + y) = f(x) + f(y)$ and $f(a x) = a f(x)$ for all $a$

This is a useful name, because it is a "function" that satisfies all the requirements of being "linear" (namely the ones given above).

Your example of the form

$f(x) = a x + b$

is usually called "linear equation" instead. Notice that for non-zero values of $b$, a linear equation does not represent a linear function, but an "affine function".

Personaly, I think that mathematics requires careful and strict use of labeling and naming for mathematical objects. Arguments of the form

"Linear function" means "first-degree polynomial function"

are not helpful because you do not gain anything from it. There already are clear and precise names for the objects you are talking about ("first-degree polynomial function").

One could think of "Linear function means first-degree polynomial function" as a mathematical insight like: "Any linear function can be expressed as a first-degree polyonomial function", but that would not help you in deciding whether "A constant function is a linear function" is true.

The fact that you list "pros" and "cons" for deciding whether a mathematical object has a certain mathematical property indicates that you are missing strict definitions. Whatever you decide "linear function" means should give you a clear way of deciding whether or not an object satisfies that definition.

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    $\begingroup$ Definitions are not etched in stone or handed down by divine writ -- they are conventions that are chosen for utilitarian reasons. My question is not "What are linear functions?" but rather what the definition should be. I can certainly accept as one answer that the definition should conform with the usage of "linear" in the sense of linear algebra, but that is certainly not standard in the context of Precalculus. $\endgroup$
    – mweiss
    Jul 25, 2016 at 13:43
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    $\begingroup$ +1 for "missing strict definitions". That some precalculus materials ignore the strict definitions is harmful in that they contribute to this sort of confusion. It may be unfortunate that the strict definitions make "line" and "linear" different in a way that can be surprising, but the definitions currently accepted by essentially all mathematicians are not things that anyone can just change. $\endgroup$
    – ShadSterling
    Jul 25, 2016 at 14:25
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    $\begingroup$ @Polyergic, I don't think "strict definitions" really exist, despite our wishes that they would, so as to simplify conversation. That is, I don't think that any super-precise definition is "accepted by essentially all mathematicians", exactly because different contexts urge different senses. $\endgroup$
    – paul garrett
    Jul 26, 2016 at 0:15
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    $\begingroup$ IMO this answer misunderstands the OP. Per mweiss' and paul garrett's comments, definitions are themselves the products of mathematical negotiation and honing. (How else would they come into being?) The OP is an invitation to hone a definition together. $\endgroup$
    – benblumsmith
    Aug 11, 2016 at 16:43
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The "problem" here is that there are two similar-but-distinct uses. It's more than the uses distinguished by the French usage noted by @Alexey. I think (at least) the English contemporary use of "linear" (or "quadratic", and others) is genuinely ambiguous, as it may refer to both polynomials, or to functions. Yes, polynomials also give functions, and we teach beginners to effectively identify the two, also.

As to polynomials, it would be unfortunate if the sum of two linear polynomials (or any linear combination, ahem) were not guaranteed to be linear, again. Or any linear combination of degree $n$ (-or-less, ahem) polynomials again considered to be of that degree. That is, shouldn't these form vector spaces?

On the other hand, "linear functions" should be ... um... linear maps between vector spaces, probably. (Here we might also have affine as a possible legitimate modifier.)

Except that, yes, with the identification of polynomials with functions, $x\to ax+b$ is "linear".

"In real life", one might feel the need to set the context by adding modifiers like "homogeneous" (of polynomials) or "non-zero homogeneous" (of polynomials). Linear maps (between two fixed vector spaces) can be added, but also composed ($X\to Y\to Z$).

And, yes, also, as Benoit Kloekner commented, the question of whether "a linear (map/function? polynomial?) is invertible" probably has no well-defined answer without clarifying the context.

EDIT: that is, to be clear, in English especially, there is genuine ambiguity in usage. Thus, to base a question (to absolute novices) on the pretense of somehow dismissing this ambiguity is not so good, not useful, not constructive. That is, the quiz-question in the question above is not reasonably answerable as "yes" or "no"... I claim.

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  • $\begingroup$ That is, without context, there is no guaranteed way to know what any author means by "linear". A question about good usage has the answer that it is inevitably ambiguous, I think. A question about "most common use", including all the elementary textbooks, may yield a different answer, but given the high auto-correlation (and lack of serious mathematical, as opposed to publishers', rationales for choices there), I am not terribly moved by textbook popularity. Similarly, this is the sort of issue that Wiki does not reliably manage wisely, for the same reasons... $\endgroup$
    – paul garrett
    Jul 25, 2016 at 20:52
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For a lateral answer... the term "linear polynomial" unambiguously excludes the constant polynomials.

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    $\begingroup$ I'd disagree, actually, ... I think the point is that although a certain fraction of the population would agree with you, another (non-trivial) would not. Thus, we cannot reason with certitude about any particular (random) usage. $\endgroup$
    – paul garrett
    Jul 26, 2016 at 0:13

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