The function $f$ has the following properties:

(i) its domain is all real numbers,

(ii) it is an odd function (that is, $f(-x)=-f(x)$ for every real number $x$ ), and

(iii) $f(2 x-3)-2 f(3 x-10)+f(x-3)=28-6 x$ for every real number $x$.

Determine the value of $f(4)$.
Since $f(2 x-3)-2 f(3 x-10)+f(x-3)=28-6 x$ for all real numbers $x$, then when $x=2$, we obtain $f(2(2)-3)-2 f(3(2)-10)+f(2-3)=28-6(2)$ and so $f(1)-2 f(-4)+f(-1)=16$. Since $f$ is an odd function, then $f(-1)=-f(1)$ or $f(1)+f(-1)=0$.

Combining with $f(1)-2 f(-4)+f(-1)=16$, we obtain $-2 f(-4)=16$ and so $-f(-4)=8$. Since $f$ is an odd function, then $f(4)=-f(-4)=8$.

ANSWER: 8