For the function $f$, what is $f(2)$ A. 6 B. 3 C. 4 D. DNE

Step 1 ：Since $f(x)$ is a linear function, we can write $f(x) = ax + b$.

Step 2 ：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 $f(g(x))=ag(x)+b.$

Step 3 ：Using that the left side is $f(g(x))=x$ we get $x=ag(x)+b.$

Step 4 ：Solving for $g$ we obtain $g(x)=\frac{x-b}{a}.$

Step 5 ：Substituting $f(x)$ and $g(x)$ into the given equation, we get $ax + b = 4 \cdot \frac{x-b}{a} + 6$

Step 6 ：Multiplying both sides by $a$, we get $a^2 x + ab = 4x - 4b + 6a.$

Step 7 ：For this equation to hold for 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 $a = \pm2$. 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 $f(x) = -2x - 6$.

Step 8 ：We're given that $f(1) = 4$, and testing this shows that the first function is the correct choice.

Step 9 ：So finally, $f(2) = 2(2) + 2 = \boxed{6}$.