Question

已知f((1+m)/m)=(㎡+1)/(㎡+1/m) 求f(m),急谢谢大家

Answer 1

f((1+m)/m) = (m²+1）/(m²+1/m)
f(1/m + 1) = [1 + (1/m)²]/[1 + (1/m)³]
设 x = 1/m + 1，则 1/m = x - 1
那么，上式可以变换成为：
f(x) = [1 + (x-1)²]/[1 + (x-1)³]
= (1 + x² - 2x + 1)/(1 + x³ - 3x² + 3x - 1)
= (x² - 2x + 2)/(x³ - 3x² + 3x)
所以，
f(m) = (m² - 2x + 2)/(m³ - 3m² + 3m)

追问

谢谢！

Answer 2

f(1+1/m)=(㎡+1)/(㎡+1/m)
设t=1+1/m,则m=1/(t-1)
f(t)=[1/(t-1)²+1]/[1/(t-1)²+t-1]
=[1+(t-1)²]/[1+(t-1)³]
所以
f(m)=[1+(m-1)²]/[1+(m-1)³]
=(m²-2m+2)/(m³-3m²+3m)