Question

The function $y=f\left(x\right)$ satisfying the condition $f(x+1/x)=x^3+1/x^3$ is?

• A. $f\left(x\right)=x^2$
• B. $f\left(x\right)=x^2-2$
• C. $f\left(x\right)=x^2+2$
• D. $f\left(x\right)=x^3-3x$

Answer 1

The correct option is D $f\left(x\right)=x^3-3x$

If $f\left(x\right)=x^2,f(x+\frac{1}{x})={(x+\frac{1}{x})}^2=x^2+2+\frac{1}{x^2}$

If $f\left(x\right)=x^2+2,f(x+\frac{1}{x})={(x+\frac{1}{x})}^2+2=x^2+4+\frac{1}{x^2}$

If $f\left(x\right)=x^2-2,f(x+\frac{1}{x})={(x+\frac{1}{x})}^2-2=x^2+\frac{1}{x^2}$

If $f\left(x\right)=x^3-3x,f(x+\frac{1}{x})={(x+\frac{1}{x})}^3-3(x+\frac{1}{x})=x^3+3x+3\frac{1}{x}+\frac{1}{x^3}-3(x+\frac{1}{x})=x^3+\frac{1}{x^3}$

This is the required solution.