is there any maping from $S^1$ to $S^1$ of odd degree, which is not an odd function?

Question

I want to prove for every continous function from $s^n$ to $s^n$ of odd degree there exists $x$ such that $f(-x)=-f(x)$ so I used this "that the sum of two functions of odd degree must be odd" but I don't know if this is really true or not. I just know that sum of two odd functions is odd but I don't think every function of odd degree has to be an odd function, so I'm trying to find an example for this using a function from $s^1$ to $s^1$. thanks for any help

Answer 1

$$(\cos\theta,\sin\theta) \mapsto (\cos(3\theta+\alpha), \sin(3\theta+\alpha)),$$ where $\alpha$ is some small angle, winds three times around the circle every time the argument to the function goes around once, but it's not an odd function.

$$ (\cos\theta,\sin\theta) \mapsto\left(\cos\left( 6\pi\left(\frac\theta{2\pi} \right)^2\right), \sin\left( 6\pi\left( \frac\theta{2\pi} \right)^2 \right) \right) \quad \text{for } 0\le\theta< 2\pi $$ also winds around three times , but it's very far from being an odd function.