Question

If $f:R\to R$, then $f\left(x\right)=(x-1)(x-2)(x-3)$ is

• A. surjective but not injective
• B. injective but not surjective
• C. both injective and surjective
• D. neither injective nor surjective

Answer 1

The correct option is A surjective but not injective


Graphically, $f\left(x\right)=(x-1)(x-2)(x-3)$
by horizontal line test, clearly many-one as it will cut more than one point.
Also, for $x\in R,$ range will be $R$
Therefore, onto function.

Alternate Solution
$\because f\left(1\right)=f\left(2\right)=f\left(3\right).$ So, $f$ is not injective.
As we know that, range for polynomial of odd degree $=R=$ Co-domain
So, $f$ is surjective.