Q48 / 120 · VITEEE 2012 · Mathematics

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

  • A
    one-one but not onto
  • Correct answer
    B
    onto but not one-one
  • C
    both one-one and onto
  • D
    neither one-one nor on to

Solution

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

Hence, the correct answer is option: 'B'.

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