Question

$f:R\to R$ and $f\left(x\right)=\frac{x(x^4+1)(x+1)+x^4+2}{x^2+x+1}$, then $f\left(x\right)$ is

• A. one-one ito
• B. many-one onto
• C. one-one onto
• D. many-one into

Answer 1

Answer: (D)

many-one into

$f\left(x\right)=\frac{x(x^4+1)(x+1)+x^4+2}{x^2+x+1}$
$=\frac{x(x^4+1)(x+1)+(x^4+1)+1}{x^2+x+1}$
$=\frac{(x^4+1)(x^2+x+1)+1}{x^2+x+1}$
$=(x^4+1)+\frac{1}{x^2+x+1}$
$f^{\prime }\left(x\right)=4x^3-\frac{2x+1}{(x^2+x+1{)}^2}=$ not always positive or negative
Thus, $f$ is many one.

Also range and co-domain of $f$ are not same,
Hence is many-one into function