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What are some common ways students get confused about finding an inverse of a function?

One I can think of is conflating multiplicative inverses of rational numbers with functional inverses. e.g. thinking "the inverse of the function $f(x)=x$ is $f^{^{-1}}(x)=\frac{1}{x}$."

I'm inexperienced and I'm trying to anticipate the ways which someone would incorrectly find an inverse and what thinking motivates it.

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    $\begingroup$ Lots of confusion also exists for some students wrt $\sin^{-1} x$, when they equate it with $\frac 1{\sin x}$, and ditto for other inverse trig functions. But that is another example of conflating multiplicative inverses with functional inverses. $\endgroup$ – Namaste May 3 at 15:50
  • $\begingroup$ @Namaste $\sin^{-1}$ and the other trig functions cause the least confusion. Read more about it in the wikipedia article. $\endgroup$ – llllllllllllllllllllllllllllll May 3 at 16:36
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    $\begingroup$ @BPP - The last line of your linked article is counter to the convention of the texts I am familiar with. Wiki suggest the capital letter somehow signifies inverse function as compare to the lower case, when that -1 superscript appears. The texts in my HS agree that the capital letter means the restricted range to "principal value". If this is a regional difference, I'll have to accept that, but after the last 3 weeks of helping students with this very issue, I'd be inclined to ask Namaste when she started reading my mind. $\endgroup$ – JTP - Apologise to Monica May 3 at 18:07
  • $\begingroup$ Update - sorry, the very last line referencing ISO 80000-2 attempts to clarify, and renders the notation from these texts all obsolete. Wow. Why was I not informed of this? $\endgroup$ – JTP - Apologise to Monica May 3 at 18:09
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    $\begingroup$ @BPP At about seventh or eighth grade I remember asking if $\sin^{-1} = 1/\sin$. $\endgroup$ – Tommi May 3 at 18:21
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I've noticed a few issues when students solve problems of the form, "Find the inverse of this function", and not all of the issues are necessarily because of the students' misunderstanding of what an inverse function means!

  1. Misunderstanding/forgetting the "one input $\to$ one output" defining feature of a function. This issue arises because implicitly-defined curves do have "inverses", they just may not be the graph of a function. For example, you can use desmos to plot something like $f(x)=x^3-4x$, and then if you type "$x=f(y)$", it will plot the inverse relation for you. The student has to know that this is not the graph of a function (because it fails the "vertical line test").
  2. Algebraic troubles when solving for the inverse. A good example of this is something like $f(x)=\frac{x+1}{2x-3}$. Even if students are following the instructions "set $f(x)=y$, swap all instances $x\leftrightarrow y$, they often will correctly set up $x=\frac{y+1}{2y-3}$ and then, unfortunately, have no idea how to start solving for $y$ in terms of $x$. (I believe it's the multiple instances of $y$ that seem daunting. But I've seen this happen with even simpler examples like $y=\frac{1}{\sqrt{2x-1}}$.)
  3. "Left-inverse" vs. "right-inverse" and colloquial understanding of "inverse". The best example of this is $f(x)=x^2$ and $g(x)=\sqrt{x}$. Many students intuit that these functions are "inverses" of each other because that is how they have been used in practice to solve equations. However, this glosses over the fact that $f(g(x))=x$ for all $x$, and yet $g(f(x))=|x|$ which is not necessarily $x$.
  4. Inverse vs. reciprocal. This was mentioned in the comments but it's worth reiterating. The issue goes beyond merely notation, e.g. $\arcsin(x)=\sin^{-1}(x)\neq \frac{1}{\sin(x)}$. I've even seen this notion appear when students think of the inverse of $\ln(x)$ means to "divide by ln"! (Yes, I have genuinely seen students circle an answer of $x=5/\ln$, blissfully unaware that it makes no sense.)
  5. Inability to see the relationship between the graphs of $f$ and $f^{-1}$. I don't know how to alleviate this problem, but I genuinely find it challenging to explain (and get students to understand and subsequently remember) that their graphs are mirror images across $y=x$. We look at multiple examples, we use a table of data points, we see that $(a,b)$ on $f$ becomes $(b,a)$ on $f^{-1}$ ... I think it's just a challenging concept.

For all of these reasons, I would recommend against having multiple choice questions that test (or "trap") students on these issues. These concepts are subtle and tricky, and it would be better to be able to see student work on a problem about these topics. (If you are required to use multiple choice questions by your institution, I would try to break the question down as much as possible so that you are asking about a specific step in the process of solving for an inverse function, instead of essentially giving them $f$ and asking them for $f^{-1}$ overall.)

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From a comment by the OP:

I'm trying to come up with "plausible" wrong answers for a multiple choice question about finding inverses.

Per an answer given to this question, you might be able to collect data on your students' possible answers by giving them a fill-in-the-blank quiz on inverse functions. Then, keep track of the most-common wrong answers by frequency.

In my experience, I've asked questions like:

"Suppose the function $f$ is invertible, and $f(3) = -4$. Find the value of $f^{-1}(-4)$."

Some common wrong answers (in no particular order) are:

  • $-3$
  • $4$
  • $-\frac{1}{3}$

I do not know for sure what motivates these answers, but if I had to guess, I would say the answer of $-3$ suggests the student correctly remembered something about switching input and output, but applied the "exponent" as a factor. The answer of $4$ suggests the student didn't remember the input-output switch, but used the "exponent" as a factor. The answer of $-\frac{1}{3}$ suggests the student knows the action of raising a number to the $-1$ exponent, but misinterpreted the notation for an inverse function.

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    $\begingroup$ Should you have "The answer of $4$ suggests..." rather than the same with $-4$? $\endgroup$ – Tommi May 3 at 18:20
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The inverse (or reciprocal) of a bijection $f~:~E\longrightarrow F$ (denoted $\stackrel{-1}{f}$ or $\stackrel{①}{f}$ or $f^{-1}$; the last notation is the worst and the number inside the circle should be $-1$) satisfy $\stackrel{-1}{f}\circ f=f\circ \stackrel{-1}{f}=\mathrm{id}_E=1_E$.
$f(x)$ is not a function; it is an element of $F$ and its inverse (if it exists) is $\left(f(x)\right)^{-1}=\frac{1}{f(x)}$.

You have the same word means two different things: the inverse of a function and the inverse of a real number. The same "problem" occurs in other words: compare a root of the polynomial P, a root of the equation... and a root of 4. To avoid the confusion, just state the question clearly.

  • Find the inverse of the function $f$ defined over $\mathbb{R}$ by $f(x)=x^3$. $\color{green}{Good}$
  • Find the inverse of the function $f(x)=x^3$. $\color{red}{Bad}$
  • Find the inverse of $f(x)=x^3$. $\color{red}{Very~Bad}$
  • Find the inverse of $x^3$. $\color{red}{Horrible}$
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    $\begingroup$ I'm actually not trying to avoid confusion, I'm trying to anticipate it. I'm trying to come up with "plausible" wrong answers for a multiple choice question about finding inverses. The only ways I've considered that a student might answer incorrectly are: (1) conflating multiplicative inverse of an element/number with an inverse of a function, (2) conflating additive inverse of an element/number with an inverse of a function, e.g. the inverse of $f(x)=x$ is $f^{-1}(x)=-x$, (3) dropping a $(-)$ sign or transcribing the wrong coefficients. I was curious about other peoples ideas/observations. $\endgroup$ – Eleanor Hankins May 3 at 17:08
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    $\begingroup$ also, I'm in a precalculus/algebra class so we don't use more sophisticated definitions of sets, maps, bijections, fuktors, or kategories. We use the 'graph of the function' and the 'function' interchangeable and when I was saying "functional inverse" I meant "functional" at that secondary-ed level of what a "function" is. You could be more precise and say they are finding the "graph of the inverse", you know $f(x)=y$ has an inverse $g(y)=x$ and swapping the $x$ and $y$ in $g$ gives us another function with it's equation denoted $g(x)$, whose graph is the inverse of the graph of $f(x)$. $\endgroup$ – Eleanor Hankins May 3 at 17:39
  • $\begingroup$ @EleanorHankins Emphasizing the distinction between a function $f$ and the image of $x$ ($f(x)$) should be done at the beginning. We usually warn the students about this distinction in high school way before explaining bijections, inverse functions, etc. $\endgroup$ – llllllllllllllllllllllllllllll May 3 at 17:50
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    $\begingroup$ there isn't one definitive pedagogical methodology that is "right". You have your opinion, my university has theirs. I work for them. So I'm asking for help with a thinking of student mistakes within the framework of this pedagogical variety. $\endgroup$ – Eleanor Hankins May 3 at 17:55

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