Question

f(x+1/x)=x^3+1/x^3，求f(x)的解析式

f(x+1/x)=x^3+1/x^3，求f(x)的解析式，小弟没分了，请大哥大姐谅解下，帮下忙

Answer 1

a=x+1/x
则a²=x²+2+1/x²
x²+1/x²=a²-2

x³+1/x³=(x+1/x)(x²-1+1/x²)=a(a²-2-1)
所以f(a)=a³-3a
即f(x)=x³-3x

Answer 2

因为f(x+1/x)=x^3+1/x^3
=(x+1/x)(x^2-1+1/x^2)
=(x+1/x)[(x+1/x)^2-3]，
所以 f(x)=x(x^2-3)。