Question

已知函数f(x)=a+(3/x-b)与函数g(x)=1+(c/2x+1)互为反函数,求a,b,c的值

Answer 1

y=g(x)=1+c/(2x+1)
反函数时x=1+c/(2y+1)
c/(2y+1)=x-1
2y+1=c/(x-1)
2y=c/(x-1)-1=(c-x+1)/(x-1)
y=(-x/2+c/2+1/2)/(x-1)
即f(x)=a+3/(x-b)=(ax-ab+3)/(x-b)=(-x/2+c/2+1/2)/(x-1)
a=-1/2
-ab+3=c/2+1/2
b=1
则(c+1)/2=7/2
c=6

所以a=-1/2,b=1,c=6

Answer 2

设A(a,k/a)
B(b,k/b)
则P(a,k/b)
P时中点则k/a=2k/b
b=2a
所以PA=k/a-k/b=k/a
PB=b-a=a
所以面积=k/a*a÷2=3
k=6