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Finding an inverse function

There is a procedure that people use and teach students to use for finding the inverse of a function. It goes like this (this image comes from page 10 of this document from Edexcel, but this pic is from Jo Morgan’s blog where I first saw it):

My problem with this is that it doesn’t make any sense, in two ways.

Firstly, why should this produce the inverse function at all? Why do you switch x and y? Why do you make y the subject? Why can you just declare “f-1=” at the end? I have seen any number of students in my drop-in centre who can do this, but have no idea what is going on or indeed what an inverse even is. Others know there is a procedure but struggle to remember what step comes next.

Secondly, the written working that goes with this procedure contains things that make no sense mathematically.

inverse-func-method1-fixed

In this example, you write down y=3x+2 and then x=3y+2. I’d be wondering how it follows that if y is 3x+2 then you know also that x=3y+2. Indeed if both of these are true, then wouldn’t that mean you could sub one into the other to get that y = 3(3y+2) + 2? Which would mean that y = 9y + 8, and so y=-1. But wait, where did the function go? At the end, the “f-1(x)=” appears by magic. It seems to be replacing the y, but wasn’t y = 3x+2 at the start? This isn’t good maths writing!

I use a different procedure. It has one or two extra lines, but it makes more sense mathematically (to me anyway). Moreover, it just uses one fundamental maths problem-solving strategy, rather than being a procedure per se.

Here’s an example to show you how it goes:

inverse-func-method2

There are two important things I want to say about this:

Firstly, my working includes the mathematical reasoning: the phrases with “let”, “then”, “but” and “so”. To me, this helps to make it more like proper maths writing. Also, I think it has a better chance of interpreted as reasoning rather than symbol-shuffling.

Secondly, the key strategy here is to give the inverse function a name “y” so that you can more easily reason about it in the hope to find out more explicitly what it is. This is a common maths problem-solving strategy and I think it is worth reinforcing much more than the inverse function procedure because it can be used elsewhere too! In order to do the reasoning, I have used the fundamental property of what the inverse function actually does, rather than just “switch x and y”.

If I had to write it as a procedure, this is what I would do:

  1. Give the function f-1(x) a name, like “y”.
  2. Do reasoning to figure out y in terms of x.
  3. You already know y=f-1(x), so you have the expression you need.
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3 Responses

  1. Stuart Reilly says:

    Why not avoid using y altogether:
    By definition
    f(f^-1(x))=x
    so
    3f^-1(x)+2=x
    Solve for f^-1(x)
    3f^-1(x)= x-2
    therefore
    f^-1(x)=1/3(x-2)
    as required

    • David Butler says:

      That’s very slick! Of course if it was a more complicated function, then I’d probably still use y to help me maintain my focus.

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