HMMT 二月 1998 · 代数 · 第 9 题
HMMT February 1998 — Algebra — Problem 9
题目详情
英文原题
- Suppose f ( x ) is a rational function such that 3 f ( ) + = x for x ≠ 0 .
Find f ( − 2 . )xx
解析
英文解析
- 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 − 12
2 f ( )
3 f ( − 2) + =21 − 1 − 1 124
⇒ 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)
Solving this system of equations { (1) , (2) } for f ( − 2) yields f ( − 2) = .672
20