Question

函数题求解

9题

Answer 1

8.f(sin²x)=cos2x+tan²x=1-2sin²x+sin²x/(1-sin²x)
∴f(x)=1-2x+x/(1-x)=(2x²-2x+1)/(1-x) (0<x<1)
9.af(x)+bf(-1/x)=sinx ①
af(-1/x)+bf(x)=sin(-1/x)=-sin(1/x) ②
①·a-②·b
(a²-b²)f(x)=asinx+bsin(1/x)
∴f(x)=[asinx+bsin(1/x)]/(a²-b²)
10.f(x)=e^x²
f[φ(x)]=e^(φ(x)²)=1-x
两边取ln:
φ(x)²=ln(1-x)
∵φ(x)≥0,
∴φ(x)=√[ln(1-x)]
∵φ(x)=√ln(1-x)≥0
∴定义域1-x≥1→x≤1.
11.g(x)=e^x>0
∴f[g(x)]=1 x<0
f[g(x)]=0 x=0
f[g(x)]=-1 x>0
g(f(x))=e |x|<1
g(f(x))=0 |x|=1
g(f(x))=1/e |x|>1