Question

Let $f:R\to R$ be a function such that $f\left(x\right)=x^3-6x^2+11x-6$. Then

• A. $f$ is one-one and into
• B. $f$ is many-one and into
• C. $f$ is one-one and onto
• D. $f$ is many-one and onto

Answer 1

Answer: (D)

$f$ is many-one and onto

Note that for $y=0$ we have $f\left(x\right)=x^3-6x^2+11x-6=0$.
$\therefore \left(x-1\right)\left(x-2\right)\left(x-3\right)=0$
or we have $3$ pre-images namely $x=1,2,3$ for $y=0$.
Hence the function $f\left(x\right)$ is a many one function.
Since the given function is a cubic equation, for every $y$ there exist a real number $x$ such that $y=\left(x-1\right)\left(x-2\right)\left(x-3\right)$
Hence the function is onto function.