Multiplying SL(2,R) Matrices in Iwasawa Coorindates

Question

Every matrix $g \in $ SL$_2(\mathbb{R})$ admits an Iwasama decomposition $g = K_gA_gN_g$ with $$K_g = \begin{pmatrix} \cos(\theta) & \sin(\theta) \\-\sin(\theta) & \cos(\theta)\end{pmatrix}, \hspace{1em} A_g = \begin{pmatrix} r & 0 \\ 0 & 1/r \end{pmatrix}, \hspace{1em} N_g = \begin{pmatrix} 1 & x \\ 0 & 1\end{pmatrix},$$ with $r > 0.$ What does a product $g_1g_2$ look like in these coordinates? My intuition says that rotations and scaling should add: $$K_{g_1g_2} = \begin{pmatrix} \cos(\theta_1 + \theta_2) & \sin(\theta_1 + \theta_2) \\-\sin(\theta_1 + \theta_2) & \cos(\theta_1 + \theta_2)\end{pmatrix}$$ and $$A_{g_1g_2} = \begin{pmatrix} r_1 + r_2 & 0 \\0 & 1/r_1+r_2 \end{pmatrix}.$$ But what happens to $N_g$ and why? I have no intuiton for the "quantity" $x.$