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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 is not what the intention of the question is.

How to restate the question in a rigorous way to exclude this answer?

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    $\begingroup$ Have you tried giving this question as-is in class or on a homework to see how many students actually come up with something like this? $\endgroup$
    – Steve
    Sep 22, 2022 at 13:14
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    $\begingroup$ "it is not what the intention of the question is" So what is the intention of the question? $\endgroup$
    – JRN
    Sep 22, 2022 at 13:27
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    $\begingroup$ I used to explicitly state in the instructions that none of the functions could be the identity function. And of course this was preceded by examples in class where an identity function was sometimes used in creating or evaluating compositions, and the term "identity function" was used. Incidentally, it's best to write "an identity function" under it on the blackboard -- or whatever is being used to convey the lectures -- because some students only copy what's on the blackboard regardless of what you might additionally say verbally (this advice has an obvious generalization). $\endgroup$ Sep 22, 2022 at 14:03
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    $\begingroup$ Should not the correct but undesired answer look like f(x) = sqrt (3 + g(x)), not f(x) = sqrt (3 + x) ? $\endgroup$
    – Rusty Core
    Sep 22, 2022 at 22:19
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    $\begingroup$ Few, if any students will answer with $f(x) = x$ or $g(x) = x$, and if they do, they've shown more creativity (and probably more understanding) than most. I see no problems with those answers. $\endgroup$
    – mrf
    Sep 30, 2022 at 14:53

7 Answers 7

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Keep it simple and phrase like this: Write $y=\sqrt{3+x}$ as the composition of two functions in a non-trivial way.

Most students will not appreciate the qualification about non-trivial and go on to give a good answer. If a student does give one of the two trivial decompositions, commend them for having the mindset of a mathematician and use the opportunity to explain what is meant by non-trivial.

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    $\begingroup$ You mean "one of the two trivial decompositions"? $\endgroup$
    – awe lotta
    Oct 5, 2022 at 6:04
  • $\begingroup$ @awe lotta, yes thank you for catching this. I corrected it. $\endgroup$
    – user52817
    Oct 5, 2022 at 20:24
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There are many other compositions (e.g., translate one direction and then translate the other, scale by two factors offsetting each other) that may not reflect your intentions either. Simply put, there are too many degrees of freedom in this question if you are looking for a very specific answer. I would suggest “write this function as a composition of two functions, one of which is…” as an alternative, though this really would reduce the difficulty.

You could try to come up with other criteria to exclude certain functions, but given the implied level of this question, that could really increase the difficulty.

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It might be easier to just chat with the rare student who gives that answer. If they understand why their answer is technically correct, then it is very likely that the teaching goals have been accomplished.

Otherwise, I'd say that you shouldn't just do an exercise in formulas. Make the factorization have meaning and the quantities (inputs and outputs to $f$ and $g$) have units. You can probably cook up something geometric, maybe with small alterations to the formula.

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I suppose you could add that neither of the two functions is allowed to be the identity function. That said, students might be more confused by this restriction than helped by it.

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    $\begingroup$ A closely related alternative is to ask for $f$ and $g$ such that $f(g(x))$ equals the given function but the reverse composition, $g(f(x))$ does not. It is typical when this material is covered to give exercises where students are asked to compute $f(g(x))$, $g(f(x))$, $f(f(x))$, etc. for given functions. So they will already be familiar with the idea that order of composition matters. If, as Dave Renfro suggests in a comment, students have been provided practice with the situation where one function is the identity function the import of the instruction should be reasonably clear to them. $\endgroup$ Sep 23, 2022 at 13:51
  • $\begingroup$ @WillOrrick, that's a good point! $\endgroup$
    – J W
    Sep 23, 2022 at 14:18
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    $\begingroup$ If their knowledge is shaky then they may actually think there's some mathematical reason, perhaps definitional, that forbids the identity function, rather than merely teacher preference. $\endgroup$
    – Thierry
    Sep 30, 2022 at 22:07
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    $\begingroup$ @Thierry Most likely a list of such problems will be assigned. You can point out to students that if the identity function is allowed, the sequence of exercises becomes pointless: every single one can be answered by writing $f(x)=x$, $g(x)=[\text{given function}]$. No skills will be learned that way. Even shaky students can appreciate teacher-imposed restrictions needed to preserve the pedagogical value of the assignment. $\endgroup$ Oct 1, 2022 at 14:51
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If this is an exercise from an introductory calculus course, one place students are going to need this skill is in applying the chain rule. The alternative composition you propose, as well as the alternative compositions in Steve's answer are not going to be very helpful in that context.

For the purposes of the chain rule, what is needed is a decomposition into simple functions, each of which the student knows how to differentiate. Since students usually learn rules for derivatives of polynomial functions, power functions, the logarithmic and exponential functions, and trigonometric functions, you could ask for a decomposition into only functions from that list. You need to make sure that students will recognize the square-root function as a power function (a skill that, again, will be required when it comes time to learn differentiation).

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IMHO, students often struggle with concepts if they don't see their usefulness. So, embedding the question into a context like "to help finding the first derivative using the chain rule" will guide the students into the desired direction.

If this isn't possible within your curriculum, you have a didactic problem, teaching something apparently useless to your students. Alas, this happens all too often in curricula.

If you can't embed the exercise into a useful context, you can resort to a clarification like

Write $y=\sqrt{3+x}$ as the composite of two functions. A solution using $f(x)=x$ or $g(x)=x$ does not count as a valid answer.

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  • $\begingroup$ I completely agree with the sentiment, but not with your precise formulation. In particular, the first sentence asks for "two functions" without giving them names, and in the second sentence you're referring to a function called $f$ and a function called $g$ that haven't been introduced before. (And in fact $f = g$, so it's very unclear and confusing why this function has two different names). $\endgroup$
    – Stef
    Oct 6, 2022 at 10:22
  • $\begingroup$ I suggest: "Find two functions $f$ and $g$ such that $\sqrt{3+x} = g(f(x))$, and such that neither $f$ nor $g$ is equal to the identity function $\operatorname{id}$ defined by $\operatorname{id}(x) = x$" $\endgroup$
    – Stef
    Oct 6, 2022 at 10:24
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What if you asked to express the given function as a composite function of two other non-composite functions. In this case you can not use the given function again because it is composite.

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  • $\begingroup$ I don't understand what you mean by a "non-composite function". Almost all functions can be written as a (non-trivial) composition of two other functions. For instance, the function $x \mapsto \sqrt{x}$ on $[0,\infty)$ can be written as a composition of $x \mapsto x^{1/4}$ and $x \mapsto x^2$, so it doesn't seem to make sense to call the square root function "non-composite". $\endgroup$ Oct 28, 2022 at 11:52
  • $\begingroup$ @JochenGlueck I meant x or x+3 , can you say those are also composite. $\endgroup$ Oct 28, 2022 at 12:01
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    $\begingroup$ Well, $x \mapsto x+3$ ist for instance the composition of $x\mapsto x+1$ and $x \mapsto x+2$. And $x \mapsto x$ is, for instance, the composition of $x \mapsto x^5$ and $x \mapsto x^{1/5}$. As I said, almost every function can be written as a non-trivial composition of two other functions. $\endgroup$ Oct 28, 2022 at 12:07
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    $\begingroup$ @JochenGlueck ok I got it, then even x can be composite function of x+1 and x -1 . $\endgroup$ Oct 28, 2022 at 12:56

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