若a>0,a≠1.F(x)为偶函数,则G(X)=F(X)×loga(x+根号下x^2+1)的图象关于原点对称.why?

G(X)=F(X)×loga[x+√(x²+1)],F(x)=F(-x)
G(-x)=F(-x)×loga[-x+√(x²+1)]=F(x)×loga[-x+√(x²+1)]
那么G(x)+G(-x)=F(x)×{loga[x+√(x²+1)]+loga[-x+√(x²+1)]}
=F(x)×loga{[x+√(x²+1)][-x+√(x²+1)]}
=F(x)×loga[(x²+1)-x²]
=F(x)×loga1
=0
那么G(x)=-G(-x),所以G(x)是奇函数,那么G(x)的图象关于原点对称

G(x)=F(x)loga[x+√(x²+1)]
G(-x)=F(-x)loga[-x+√((-x)²+1)] //F(x)=F(-x)
=F(x)loga[√(x²+1)-x]
=F(x)loga{[√(x²+1)-x][√(x²+1)+x]/[√(x²+1)+x]}
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