Question

求函数f(x)的解析式

Answer 1

2x * f(x) + (x - 3) * f(1/(1-x)) = 4x² - 10x - 1/2 ... (1) 令 y = 1/(1-x)
则 1/(1-y) = (x-1)/x = z
那么 1/(1-z) = x。 分别以y
z 代入(1) 得 :2x * f(x) + (x - 3) * f(y) = 4x² - 10x - 1/2 ... (1)2y * f(y) + (y - 3) * f(z) = 4y² - 10y - 1/2 ... (2)2z * f(z) + (z - 3) * f(x) = 4z² - 10z - 1/2 ... (3) (1) * 2y - (2) * (x - 3) 得 : 4xy * f(x) - (x - 3)(y - 3) * f(z) = 2y(4x² - 10x - 1/2) - (x - 3)(4y² - 10y - 1/2) ...(4) (4) * 2z + (3) * (x - 3)(y - 3) 得 :8xyz * f(x) + (x - 3)(y - 3)(z - 3) * f(x) = yz(16x² - 40x - 2) - z(x - 3)(8y² - 20y - 1) + (x - 3)(y - 3)(4z² - 10z - 1/2) 注意 y = 1/(1-x)
z = (x-1)/x
xyz = - 1
xy = x/(1-x)
yz = - 1/x
zx = x - 1 。 - 8 * f(x) + (x - 3) (2-3x)/(x-1) (-2x-1)/x * f(x) = (- 16x + 40 + 2/x) - (x - 1 - 3(x-1)/x)(8/(1-x)² - 20/(1-x) - 1) + (x - 3) (2-3x)/(x-1) (4(x-1)²/x² - 10(x-1)/x - 1/2) (6x³ - 27x² + 9x + 6) / (x(x-1)) * f(x) = (9x⁴- 93x³ + 180x² - 18x - 48) / (2x²(x - 1)) f(x) = (3x⁴- 31x³ + 60x² - 6x - 16) / (4x⁴- 18x³ + 6x² + 4x) 2012-11-15 20:59:59 补充： 不肯定~我觉得只有这个解~ 那个网我看不懂
不过你小心打上连结会被系统扣分~
How e your mand is not deleted?? I got few hundred points deleted because of posting links. Stupid YAHOO server.
雨后阳光，你咁肯定这条函数方程只有这个解？ 不怕好似f(1/x) = xf(x)（math.stackexchange/questions/28737）咁样有无限多个解，且任意部分为某种类的函数？