Question

$f:(-\infty ,\infty )\to (-\infty ,\infty )$ is defined by $f\left(x\right)=x(x-1)(x-2)$. Then $f$ is

• A. one-one but not onto
• B. onto but not one-one
• C. bijective
• D. Identity

Answer 1

The correct option is B onto but not one-one

$f:(-\infty ,\infty )\to (-\infty ,\infty )$ for $f\left(x\right)=f\left(y\right)=x(x-1)(x-2)$ has
more than one image for $f\left(0\right)=0$ viz. $x=0,x=1,x=2$
Therefore, $f$ is not one to one.
On the other hand range of $f$ is equal to co-domain of $f$.
Hence, $f$ is onto