Q41 / 160 · AP EAMCET 2024 (18 May Shift 1) · Mathematics

If a function $f: \mathrm{R} \rightarrow \mathrm{R}$ is defined by $f(x)=x^3-x$, then $f$ is

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

Solution

Given $f: \mathrm{R} \rightarrow \mathrm{R}$ such that $f(x)=x^3-x=x(x-1)(x+1)$ $\because f(1)=0=f(0)$. So, $f(x)$ is not one-one Since, $f(x)=x^3-x$ is a polynomial function So it is continuous on R and, If $x \rightarrow \infty \Rightarrow f(x) \rightarrow \infty$ and $x \rightarrow-\infty \Rightarrow f(x) \rightarrow-\infty$. So, range of $f(x)$ is $(-\infty, \infty)$ $\Rightarrow f(x)$ is an onto function

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

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