# complex analysis – Checking whether an analytic function is onto or not

Let $$f:mathbb Crightarrow mathbb C$$ be an entire function such that the function $$g(z)$$ given by $$g(z)=f(frac{1}{z})$$ has a pole at 0. Prove or disprove $$f$$ is onto.

I think $$f$$ is an onto function.

My attempt:
Since $$f$$ is an entire function $$f$$ has a power series representation about 0 given by $$sum_{m=0}^infty a_m z^m$$. Then $$g(z)=sum_{m=0}^infty frac {a_m}{z^m}$$. Since $$g$$ has pole at 0 then $$g$$ is of the form $$g(z)=a_0+frac{a_1}{z}+frac{a_2}{z^2}+…+frac{a_n}{z^n}$$ for some fixed $$nin mathbb N$$. Then $$f(z)=a_0+a_1z+a_2z^2+….+a_nz^n$$. Since $$f$$ is a polynomial it follows from Fundamental Theorem of Algebra that $$f$$ is onto.