Re: [中學] 函數方程
※ 引述《jurian0101 (Hysterisis)》之銘言：
: 從筆記堆中挖出來的老題目，頗奇妙的
: f(x) + f((x-3)/(x-2)) = x-1
: 試解出f(x)之顯式
denote u = (x-3)/(x-2)
(u-3)/(u-2) = [(x-3-3x+6)/(x-2)]/[(x-3-2x+4)/(x-2)]
= (2x-3)/(x-1)
denote v = (2x-3)/(x-1)
(v-3)/(v-2) = [(2x-3-3x+3)/(x-1)]/[(2x-3-2x+2)/(x-1)]
= x
f(x) + f(u) = x-1
f(u) + f(v) = u-1
f(v) + f(x) = v-1
2f(x) = x+v-u-1
f(x) = [x + (2x-3)/(x-1) - (x-3)/(x-2) -1] / 2
= [x + 2 - 1/(x-1) - 1 + 1/(x-2) -1] / 2
= [x - 1/(x-1) + 1/(x-2)] / 2
: - - - -
: 有關聯的附題:
: 試解出所有 f(f(f(x)))=x 的函數f
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 123.194.224.241
※ 編輯: JohnMash 來自: 123.194.224.241 (08/20 00:35)
推
08/20 01:07, , 1F
08/20 01:07, 1F
→
08/20 12:41, , 2F
08/20 12:41, 2F
→
08/20 13:13, , 3F
08/20 13:13, 3F
→
08/20 13:20, , 4F
08/20 13:20, 4F
→
08/20 13:34, , 5F
08/20 13:34, 5F
→
08/20 13:41, , 6F
08/20 13:41, 6F
→
08/20 21:37, , 7F
08/20 21:37, 7F
討論串 (同標題文章)