Track / Stage 6 / 182 of 400 #1182 of 1964

Problem 1182

National olympiad, first round

Algebra Difficulty 6.3 Find the answer

137. 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 $f_{-1}(x)$ .

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Official solution

137. The scheme of the function $\frac{4 x+1}{2 x+3}$ is shown in Fig. 149. It consists of two fixed points and three cycles.
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Fig. 149.
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Fig. 150.

The scheme of the function $\frac{2 x+1}{3 x+2}$ (Fig. 150) consists of two cycles and has no fixed points.

The scheme of the function $\frac{3 x-1}{x+1}$ (Fig. 151) consists of one fixed point and one cycle. 138. The scheme for the function $f_{-1}(x)$ is obtained from the scheme of $f(x)$ by reversing the direction of all arrows. Therefore, the statement of the problem directly follows from the fact that in the scheme of any fractional-linear function, including $f_{-1}(x)$ , one and only one arrow departs from each point.

Fig. 151.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.