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

The correct option is B onto but not one – one

We have, f(x) = (x - 1)(x - 2)(x - 3)
Since, 1, 2, 3 $ϵ$ R and f(1) = 0, f (2) = 0, f(3) = 0
$\therefore$ f is many - one
Let y $ϵ$ R. f(x) = y implies (x-1) (x-2) (x-3) = y
$\Rightarrow x^3-6x^2+11x-6-y=0\Rightarrow$ y $ϵ$ R
Thus, onto