jmc

Since $f(x)$ is a linear function, we can write $f(x)=ax+b$. We want to find the inverse function $g(x)$ defined by $f(g(x))=x$ for every $x$. If we substitute $g(x)$ into the equation for $f$ we get $$f(g(x))=ag(x)+b.$$Using that the left side is $f(g(x))=x$ we get $$x=ag(x)+b.$$Solving for $g$ we obtain $$g(x)=\frac{x-b}{a}.$$Substituting $f(x)$ and $g(x)$ into the given equation, we get $$ax+b=4\cdot \frac{x-b}{a}+6$$Multiplying both sides by $a$, we get $$a^{2}x+ab=4x-4b+6a.$$For this equation to hold for $\text{all}$ values of $x$, we must have the coefficient of $x$ on both sides equal, and the two constant terms equal. Setting the coefficients of $x$ equal gives $a^2=4$, so $a=\pm 2$. Setting constant terms equal gives $ab=-4b+6a$. If $a=2$, we have $2b=-4b+12$, which gives $b=2$. If $a=-2$, we have $-2b=-4b-12$, so $b=-6$. Thus we have two possibilities: $f(x)=2x+2$ or $f(x)=-2x-6$.

We're given that $f(1)=4$, and testing this shows that the first function is the correct choice. So finally, $f(2)=2(2)+2=\boxed{6}$.