Verify that $f =f_{+1}∘f_{∗2}∘R∘f_{−1} $.

Visualize $f$ with a mapping diagram that illustrates the function as the composition: $f = f_{+1}∘f_{∗2}∘R∘f_{−1}$.

Draw a mapping diagram yourself or consider the GeoGebra figures below.

Given a point / number, $x$, on the source line, there is a unique arrow meeting the target line at the point / number, $\frac 2 {x-1} + 1 = \frac {x + 1} {x-1}$, which corresponds to the function's value for $x$

When the point in the domain is $0$, the black arrow points to $f(0) = -1$ visualizing the point $(0,-1)$ on the graph of $f$. We also have $f_{+1} \circ f_{*2} \circ R\circ f_{ -1} (2) = f_{+1} \circ f_{*2}\circ R(1)= f_{+1}(2) = 3$ which is visualized in the mapping diagram for the composition by arrows $2 \rightarrow 1 \rightarrow 1 \rightarrow 2 \rightarrow 3$ .