Find a period if $f(x+k)=1+(2-5f(x)+10f(x)^2-10f(x)^3+5f(x)^4-f(x)^5)^{\frac{1}{5}}$

Question

Let $f$ be a real valued function with domain $\mathbb{R}$ satisfying, $$f(x+k)=1+(2-5f(x)+10f(x)^2-10f(x)^3+5f(x)^4-f(x)^5)^{\frac{1}{5}}$$ for all real $x$ and some positive constant $k$, then find the period of $f(x)$ if it is periodic.

My Attempt:

$$f(x+k)=1+(1+(C_0^5(1)^5(-f(x))^0+C_1^5(1)^4(-f(x))+C_2^5(1)^3(-f(x))^2+C_3^5(1)^2(-f(x))^3+C_4^5(1)^1(-f(x))^4+C_5^5(1)^0(-f(x))^5))^{\frac{1}{5}}$$

$$f(x+k)=1+(1+(1-f(x)^5)^{\frac{1}{5}}$$

The problem is that I am not able to establish $f(x)$ is periodic. How can I proceed ?

Answer 1

Write $g(x) = (1-f(x))^5+1/2$ and you have $$g(x+k)=-g(x)$$

so $$g(x+2k) =-g(x+k)=g(x)$$

so $g$ has period $2k$.

So $$(1-f(x+2k))^5 = (1-f(x))^5\implies 1-f(x+2k) = 1-f(x)$$ and so $f$ has also period $2k$.