Question

2f(x)+2f(1-x)=x²,求f(x)解析式

Answer 1

2f(x)+f(1-x)=x²

用(1-x)替换x

2f(1-x)+f[1-(1-x)]=(1-x)²

2f(1-x)+f(x)=(1-x)²

相加得

3f(1-x)+3f(x)=x²+(1-x)²

3[f(1-x)+f(x)]=2x²-2x+1

f(1-x)+f(x)=(2x²-2x+1)/3

f(1-x)+2f(x)=x²

相减得f(x)=x²-(2x²-2x+1)/3

f(x)=(x²+2x-1)/3