Inverse function mod $$ y = 3x + 7 \pmod 4,\quad x,y \in \mathbb{Z}_4 $$ 
$x = 2, y =$ ?? 
Find the inverse function and verify the value of $y$.
I solved only the first question $y = 13\bmod 4  = 1 \bmod 4. 
How to solve the second question for inverse?

The only possibilities are:
$$(x,y) \in \{ (0,3),(1,2),(2,1),(3,0) \}$$
So the functions are as follows:
$$y=\left\{\begin{array}{c} 3 & x=0 \\ 2 & x=1 \\ 1 & x=2 \\ 0 & x=3 \\ \end{array}\right\}$$
$$x=\left\{\begin{array}{c} 3 & y=0 \\ 2 & y=1 \\ 1 & y=2 \\ 0 & y=3 \\ \end{array}\right\}$$
And if you need some formula, it is:
$$y \overset{4}{\equiv} 3x+7 \overset{4}{\equiv} -x+3 \Longrightarrow x+y \overset{4}{\equiv} 3 \Longrightarrow \left\{ \begin{array}{c} y \overset{4}{\equiv} 3-x \\ x \overset{4}{\equiv} 3-y \\ \end{array} \right\}$$
