1. ## 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

2. 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 $f(x) = x^3 - 3x$?

3. 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 $w=f(z)$ have inverses except at the zeros of $f'(z)$, with the derivative of the inverse given by $\frac{df^{-1}}{dw}=\frac{1}{f'(z(w))}$.

4. Originally Posted by shawsend
I believe all polynomials $w=f(z)$ have inverses except at the zeros of $f'(z)$, with the derivative of the inverse given by $\frac{df^{-1}}{dw}=\frac{1}{f'(z(w))}$.
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 $f^{-1}(f(x)= x$ and $f(f^{-1}(x))= x$ for all x. Such a thing exists if and only if $f'(x)\ge 0$ for all x or $f'(x)\le 0$ for all x.