Problem 1234

Construct schemes corresponding to the 7 arithmetic fractional-linear functions

$$\frac{4 x+1}{2 x+3}, \quad \frac{2 x+1}{3 x+2}, \quad \frac{3 x-1}{x+1}$$

Consider two fractional-linear functions

$$f(x)=\frac{a x+b}{c x+d} \quad \text{and} \quad g(x)=\frac{-d x+b}{c x-a}$$

It is not hard to see that if $f(x)$ maps a number $m$ to a number $n$, i.e., $f(m)=n$, then $g(x)$ maps $n$ back to $m$, $g(n)=m$. The scheme of the function $g(x)$ is obtained from the scheme of the function $f(x)$ by simply reversing the direction of all arrows.

Such a function $g(x)$ is called the inverse of $f(x)$ (the function $f(x)$, in turn, is the inverse of $g(x)$). The function inverse to $f(x)$ will be denoted by $f_{-1}(x)$.