Problem
GEO-B3-M02-P003 A Fractional Linear Check
B. New Original Problem. On the projective line, let \(f(x)=\frac{2x-1}{x+3}\). Find the images of \(0\), \(1\), \(\infty\), and \(-3\), then prove that \(f\) preserves the cross-ratio of any four points where the expressions are defined.
C. Hint 1. First compute the special values, including the point where the denominator is zero.
D. Hint 2. To prove cross-ratio preservation, compare \(f(x_i)-f(x_j)\).
E. Full Solution.
We have \(f(0)=-\frac{1}{3}\), \(f(1)=\frac{1}{4}\). At \(x=\infty\), the image is the ratio of the leading coefficients, namely \(2\). At \(x=-3\), the denominator is zero, so \(f(-3)=\infty\).
For any \(x_i,x_j\), compute:
\[ f(x_i)-f(x_j)=\frac{(2x_i-1)(x_j+3)-(2x_j-1)(x_i+3)}{(x_i+3)(x_j+3)}=\frac{7(x_i-x_j)}{(x_i+3)(x_j+3)}. \]
When forming the cross-ratio, all factors \(7\) and \((x_i+3)\) cancel. Hence \((f(x_1)f(x_2)f(x_3)f(x_4))=(x_1x_2x_3x_4)\).
A good short task connecting the coordinate formula with the geometric term “projectivity”.