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HMMT 二月 1998 · 代数 · 第 9 题

HMMT February 1998 — Algebra — Problem 9

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Suppose f ( x ) is a rational function such that 3 f ( ) + = x for x ≠ 0 . x x Find f ( − 2 . )
解析
  1. Problem: Suppose f ( x ) is a rational function such that 3 f ( ) + = x for x 6 = 0. Find f ( − 2). x x − 1 Solution: Let x = . Then 2 − 1 2 f ( ) 1 2 3 f ( − 2) + = − 1 4 2 − 1 1 ⇒ 3 f ( − 2) − 4 f ( ) = (1) 2 4 Let x = − 2. Then − 1 2 f ( − 2) 3 f ( ) + = 4 2 − 2 − 1 ⇒ 3 f ( ) − f ( − 2) = 4 (2) 2 67 Solving this system of equations { (1) , (2) } for f ( − 2) yields f ( − 2) = . 20