Why is the domain of the hyperbolic function $\sinh x$ is symmetric about the origin?

Question

My book asked me to prove that the function $\sinh x$ is odd, but in order to be odd I must be sure that the domain of it symmetric about the origin, how can I be sure from this?

Answer 1

$$f(x)=\sinh x:=\frac{e^x-e^{-x}}2=\color{red}-\frac{e^{-x}-e^x}2=-\sinh(-x)=-f(-x)$$