Thread: Polynomials and inverses.

Let P be a polynomial of degree n.

1.) Can P have an inverse if n is even. Support your answer.

2.) Can P have an inverse if n is odd? If so, give an example. Then give an example of a polynomial of odd degree that does not have an inverse.

For 1.) I think that no P's can have an inverse if they are even because they would fail the HLT.

For 2.) I'd say yes P can have an inverse if n is odd such as the inverse of f(x) = x^3 is f^-1(x) = x^(1/3).

I cannot think of an example of a polynomial of od degree that does not have an inverse though. Please help

Originally Posted by WartonMorton

Let P be a polynomial of degree n.

1.) Can P have an inverse if n is even. Support your answer.

2.) Can P have an inverse if n is odd? If so, give an example. Then give an example of a polynomial of odd degree that does not have an inverse.

For 1.) I think that no P's can have an inverse if they are even because they would fail the HLT.

For 2.) I'd say yes P can have an inverse if n is odd such as the inverse of f(x) = x^3 is f^-1(x) = x^(1/3).

I cannot think of an example of a polynomial of od degree that does not have an inverse though. Please help

How about ?

Originally Posted by WartonMorton

Let P be a polynomial of degree n.

1.) Can P have an inverse if n is even. Support your answer.

2.) Can P have an inverse if n is odd? If so, give an example. Then give an example of a polynomial of odd degree that does not have an inverse.

For 1.) I think that no P's can have an inverse if they are even because they would fail the HLT.

For 2.) I'd say yes P can have an inverse if n is odd such as the inverse of f(x) = x^3 is f^-1(x) = x^(1/3).

I cannot think of an example of a polynomial of od degree that does not have an inverse though. Please help

I believe all polynomials have inverses except at the zeros of , with the derivative of the inverse given by .

Originally Posted by shawsend

I believe all polynomials have inverses except at the zeros of , with the derivative of the inverse given by .

Be careful about what you are calling "inverses". Every function, f, has a local inverse about a point where f'(x) is not 0. That is, the function defined by restricting f to some small neighborhood of the point has an inverse.

The problem is clearly talking about "the" inverse such that and for all x. Such a thing exists if and only if for all x or for all x.