This statement is Theorem 1.1 (page 39) of Computability, Complexity and languages by Martin Davis:
If function $h$ is obtained from the (partially) computable functions $f$, $g_1$, $g_2$, ..., $g_k$ by composition then $h$ is (partially) computable.
What I understand from this theorem is if there is at least one partial function among the composed function then $h$ is partial function. Am I right?
Here is an example, the functions $x$ and $x+y$ and $x.y$ are total but $x-y$ is a partial function. it is possible to obtain $4x^2-2x$ by composition of these functions in the following way, this function is total but it is obtained from an non-total function ($x-y$).
$2x = x+x$
$4x^2 = (2x)(2x)$
$4x^2-2x$ composition of $x-y$
This is an example from the book, is this example a contradiction of the theorem?