已知函数fx=ax+b,且f(ax+b)=4x+6,求fx的解析式
解:f(x)=ax+b
f(ax+b)=4x+6
a(ax+b)+b=4x+6
整理,得(a²-4)x+(ab+b-6)=0
要对任意实数x,等式恒成立
a²-4=0,ab+b-6=0
a²=4,a=2或a=-2
a=2时,2b+b-6=0,3b=6,b=2
a=-2时,-2b+b-6=0,b+6=0,b=-6
函数f(x)的解析式为:f(x)=2x+2或f(x)=-2x-6
写为f(u)=au+b
则f(ax+b)=a(ax+b)+b=(a^2)x+(a+1)b=4x+6
所以
a^2=4,(a+1)b=6
a=2或-2,对应的b=2或-6
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