= Solution
In the <upper half-plane model>, <hyperbolic lines> are vertical Euclidean rays and semicircles centred on the real axis, equivalently circle arcs meeting the real boundary orthogonally. A semicircle with endpoints $r<s$ is mapped to the imaginary axis by
$$
g(z)=\frac{z-r}{s-z},\qquad \text{matrix representative }\frac1{\sqrt{s-r}}\begin{pmatrix}1&-r\\-1&s\end{pmatrix}\in\operatorname{SL}(2,\mathbb R).
$$
Its determinant is one, and its real endpoints map to $0,\infty$. A vertical line $x=a$ is mapped to the imaginary axis by $z\mapsto z-a$. Thus every hyperbolic line can be carried to a fixed one; composing one such map with the inverse of another proves transitivity of $\operatorname{PSL}(2,\mathbb R)$ on the lines.
Reflection in $x=a$ is $R_a(z)=2a-\overline z$. Reflection in the unit semicircle is $R_\circ(z)=1/\overline z$. The latter fixes the semicircle pointwise, is an involution, and preserves the <hyperbolic metric> because $|dw|=|dz|/|z|^2$ and $\operatorname{Im}w=\operatorname{Im}z/|z|^2$. The vertical reflection similarly preserves $|dz|/\operatorname{Im}z$. Their composition in the specified order is
$$
\boxed{R_aR_\circ(z)=2a-\frac1z.}
$$
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