How can I find the equation of the inverse function of a cubic function?

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Discussion Overview

The discussion revolves around finding the equation of the inverse function of a specific cubic function, f(x) = x^3 + 6x^2 + 12x + 7. Participants explore various methods and concepts related to inverse functions, including algebraic techniques and theoretical implications.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Nan describes her progress in finding the inverse function, noting the inflection point and the roots of the cubic function.
  • Some participants suggest using Cardano's formula to find the roots, but there is uncertainty about how this relates to finding an explicit inverse function.
  • There is a suggestion that cubics often do not have nice, closed form inverses, raising questions about the nature of the inverse function.
  • One participant proposes rewriting the equation to isolate y, leading to a cubic equation that can be solved using Cardano's method.
  • Another participant mentions the idea of "completing the cube" as a potential method, although it is debated whether this approach is valid for cubics.
  • There is a discussion about the implications of the coefficients in the cubic function and how they affect the ability to find an inverse.
  • Nan expresses gratitude for the suggestions and indicates she feels more directed in her approach to completing her paper.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best method to find the inverse function, with multiple competing views and techniques proposed. The discussion remains unresolved regarding the explicit form of the inverse function.

Contextual Notes

Some participants note that the nature of cubic functions may complicate the existence of a simple inverse, and there are unresolved mathematical steps regarding the application of various methods discussed.

agentnan
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I am working on a paper to find the equation of the inverse function of a cubic function. The function is

f(x) = x^3 + 6x^2 +12x +7​

I have already graphed the function and its inverse. I have found the inflection point of (-2, -1). I found the 3 roots (1 real & 2 complex). The real root is x=-1, so I am left with:

(x+1)(x^2 + 5x + 7) ​

which leads to the complex roots of:

(5+i(3)^.5)/2 and (5 - i(3)^.5)/2​
.

At this point though I am totally stuck. I know that I am supposed to switch the placement of the x & y variables, but I can't figure out what the next step should be. So I now have:

x = y^3 + 6y^2 +12y +7,​

but I have no idea what to do with it! I know from above that it can be rewritten as:

x = (y+1)(y^2 + 5y + 7).​

How do I get the y variable all by itself though? Looking for some direction on what I should do at this point to come up with the equation of the inverse.

Thanks for your help in advance!
Nan
 
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You can Google for Cardano's formula... but why do you want an explicit inverse?
 
The Cardano's formula will give me the values for x which I already have. I am not sure how that formula will help me get an explicit inverse other than giving me the values for x.

The person who assigned this paper asked us to "find the inverse function of the given function",. He also stated "the purpose of theis research assignment is to test your ability to apply the concept of an inverse function and other algebraic concepts to a problem."

So I am assuming that he wants an actual equation of the inverse. We tried to ask for further details, but we were told that it was a research paper and he would not give us any help on it. Perhaps I am misreading what he is requesting? If you have any thoughts on what he may actually be asking for, it would be terrific! As mentioned, I have already graphed it, found the inflection point & the 3 roots.
Thanks again!
Nan
 
Last edited:
Well, there's always Galois theory, perhaps that could give some insight.
 
I will take a look at it in a few minutes, but I should have mentioned this is for an intermediate Algebra class, not a high level course. Thanks for the suggestion!
 
Okay...I took a quick peek at the Galois Theory...but it appears to be way beyond the Algebra 2 level. Any other thoughts on this problem?
 
agentnan said:
I am working on a paper to find the equation of the inverse function of a cubic function. The function is

f(x) = x^3 + 6x^2 +12x +7​

I have already graphed the function and its inverse. I have found the inflection point of (-2, -1). I found the 3 roots (1 real & 2 complex). The real root is x=-1, so I am left with:

(x+1)(x^2 + 5x + 7) ​

which leads to the complex roots of:

(5+i(3)^.5)/2 and (5 - i(3)^.5)/2​
.

At this point though I am totally stuck. I know that I am supposed to switch the placement of the x & y variables, but I can't figure out what the next step should be. So I now have:

x = y^3 + 6y^2 +12y +7,​

but I have no idea what to do with it! I know from above that it can be rewritten as:

x = (y+1)(y^2 + 5y + 7).​

How do I get the y variable all by itself though? Looking for some direction on what I should do at this point to come up with the equation of the inverse.

Thanks for your help in advance!
Nan

Generally speaking, cubics often do NOT have nice, closed form inverses. You are trying to find f^{-1}(x) as a nice formula in terms of x, however this form implicitly assumes that f^{-1}(x) is a function. Generally, it is not. Here's how to see this geometrically.

If you have the graph of a function f, you can easily obtain the graph of the "inverse" of by reflecting the graph of f across the line y=x. Draw a general cubic, and reflect its graph across y=x. You can find a cubic so that its inverse graph does not satisfy the vertical line test.
 
If you have

x = y^3 + 6y^2 +12y +7

then write

y^3 + 6y^2 +12y + (7-x) = 0

and solve using Cardano's method.
 
Or you could try "completing the cube", as an analogy to completing the square? If I am not mistaken, this should yield a nice algebraic expression without too much computation?
 
  • #10
"Completing the cube" doesn't work, unfortunately. For a quadratic, all you have to do to complete the square is add a constant. But a cubic has too many variables to do that. Try it.
 
  • #11
But wait! This particular cubic can be completed by adding a constant! Well, then.

To the OP: Remember that

(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3

Now, write

y^3 + 6y^2 + 12y = x - 7

and see what you can do.
 
  • #12
In general yes, but x^3 + 6x^2 + 12x + 7 = (x+2)^3-1. So one would think that the coefficients 6 and 12 were not chosen arbitrarily?
 
Last edited:
  • #13
Thank you ALL so much...you have definitely sent me in the correct direction particularly Big-T and Ben Niehoff! I will finish my paper right now! Thanks again...Nan
 
Last edited:

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