Math CentralQuandaries & Queries


Question from alireza, a student:
I want inverse function for F(x)=x^3+x


I'm going to assume that anybody who wants to invert a cubic equation, and uses that notation, has some background in math. You may be aware that there is a a "cubic formula" like the quadratic formula, which can be found in any good set of math tables (CRC, Schaum's, etc). Plugging into that will give you the answer you require but is completely opaque.

Another approach is to use hyperbolic identities. We note that

sin(3A) = 3 sin(A) - 4 sin3(A)

The sign is wrong, so we guess maybe hyperbolic sine? Bingo!

sinh(3A) = 3 sinh(A) + 4 sinh3(A)

Now, find constants K, L such that

3 sinh (A) + 4 sinh^3 (A) = L [K sinh A + (K sinh A)3]

hint: K3 is to K as 4 is to 3

Then if x = K sinh A,

sinh(3A)/L = x + x3

F(x) = sinh(3A)/L

We're now ready to invert. start with F(x).

Multiply by L and take the inverse sinh to get 3A
Divide by 3 to get A
Take the sinh and multiply by K to get x.

If we had wanted just to invert y = x^3, we could have taken
a logarithm, divided by 3, and then exponentiated to get x. The type of transformation we have found works (with appropriate constants) for any cubic of the form

y = ax(p2+x2)

and always transforms inverting the cubic into division by three. Using sines will work for

y = ax(p2 - x2).

Of course, a general cubic function has the form

ax3 + bx2 + cx + d = y .

If you have a nonzero quadratic term, you can proceed as if completing the square to get rid of it. Unwanted constant terms can be moved into "y". So we have developed a complete technique for solving cubics.


About Math Central


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
Quandaries & Queries page Home page University of Regina PIMS