Question

The function $f:R\to R$ defined by $f\left(x\right)=(x-1)(x-2)(x-3)$ is.

• A. One-one but not onto
• B. Onto but not one-one
• C. Both one-one and onto
• D. Neither one nor onto

Answer 1

The correct option is B Onto but not one-one

$f\left(x\right)=(x-1)(x-2)(x-3)$

$\Rightarrow f\left(1\right)=f\left(2\right)=f\left(3\right)=0$ But $1\ne 2\ne 3$

$\therefore f\left(x\right)$ is not one-one.

For each $y\in R$, there exists $x\in R$ such that $f\left(x\right)=y$.

$\therefore$ f is onto.
[Note that if a continuous function has many than one roots, then function is always many-one.]

Hence, $f$ is onto but not one-one.