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.