Question

The domain of function satisfying $f\left(x\right)+f\left(x^{-1}\right)=\frac{x^3+1}{x}$, is

• A. an empty set
• B. a singleton
• C. a finite set
• D. an infinite set

Answer 1

The correct option is C a finite set

$y=f\left(x\right)+f\left(x^{-1}\right)=\frac{x^3+1}{x}..\left(1\right)$
Replacing $x\to x^{-1}$
$\Rightarrow y=f\left(x\right)+f\left(x^{-1}\right)=\frac{x^3+1}{x^2}..\left(2\right)$
By $\left(2\right)-\left(1\right)$, we get
$\Rightarrow \frac{x^3+1}{x^2}-\frac{x^3+1}{x}=0$
$\Rightarrow \frac{x^3+1}{x^2}(1-x)=0\Rightarrow x=\pm 1$ which is finite set.