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已知函数f(x)=x^2/(1+x^2),试求f(1)+f(2)+f(1/2)+f(3)+f(1/3)+f(4)+f(1/4)的值!
题目详情
已知函数f(x)=x^2/(1+x^2),试求f(1)+f(2)+f(1/2)+f(3)+f(1/3)+f(4)+f(1/4)的值!
▼优质解答
答案和解析
因为f(x)+f(1/x)
=x^2/(1+x^2)+(1/x^2)/[1+(1/x^2)]
=x^2/(1+x^2)+1/(1+x^2)
=(1+x^2)/(1+x^2)
=1
所以 f(2)+f(1/2)=1 f(3)+f(1/3)=1 f(4)+f(1/4)=1
原式=f(1)+[f(2)+f(1/2)]+[f(3)+f(1/3)]+[f(4)+f(1/4)]
=f(1)+1+1+1
=1/2+3
=7/2
=x^2/(1+x^2)+(1/x^2)/[1+(1/x^2)]
=x^2/(1+x^2)+1/(1+x^2)
=(1+x^2)/(1+x^2)
=1
所以 f(2)+f(1/2)=1 f(3)+f(1/3)=1 f(4)+f(1/4)=1
原式=f(1)+[f(2)+f(1/2)]+[f(3)+f(1/3)]+[f(4)+f(1/4)]
=f(1)+1+1+1
=1/2+3
=7/2
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