Question

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

• A. one-one but not onto
• B. onto but not one-one
• C. both one-one and onto
• D. neither one-one nor onto

Answer 1

Answer: (B)

$f(x)=(x-1)(x-2)(x-3)$
$\Rightarrow f(1)=f(2)=f(3)=0$
$\therefore f(x)$ is not one-one.
For each $y\in R$, there exists $x\in R$ such that $f(x)=y$.
$\therefore f$ is onto.
Note that if a continuous function has more than one roots, then the function is always many-one.