Track / Stage 4 / 337 of 340 #597 of 1964

Problem 597

AMC 12 late, AIME early

Algebra Difficulty 5.0 Find the answer

Folklore

The function $f(x)$ is defined for all $x$ , except 1, and satisfies the equation: $(x-1) f\left(\frac{x+1}{x-1}\right)=x+f(x)$ . Find $f(-1)$ .

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

Substitute the values $x=0$ and $x=-1$ into the given equation. We get: $\left\{\begin{array}{c}-f(-1)=f(0), \\ -2 f(0)=-1+f(-1)\end{array}\right.$ . Therefore, $2 f(-1)=-1+f(-1)$ , which means $f(-1)=-1$ .

## Answer

-1.

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