I'm reading the book "Complex Variables and Applications" by Brown and Churchill. 

I'm confused about complex mappings. Say we have $f(x+i y)=u(x,y)+i v(x,y)$. 

I'm given to understand that if the mapping is represented in the form $f(z)=w$, then $x+iy$ is mapped in the $z$-plane, and $u(x,y)+iv(x,y)$ in the $w$-plane. Moreover, in the $z$-plane, $y$ is expressed as a function of $x$ ($y=g(x)$), and in the $w$-plane $v(x,y)=h(u(x,y))$. 

How is it obvious that $v(x,y)$ can be expressed as a function of $u(x,y)$?

Thanks!