Q230 / 270 · NDA 2017 (Phase 2) · Mathematics

If $f(x)=\frac{x}{2}-1$, then on the interval $[0, \pi]$ which one of the following is correct?

  • A
    $\tan [f(x)]$, where $[\cdot]$ is the greatest integer function, and $\frac{1}{f(\mathrm{x})}$ are both continuous.
  • B
    $\tan [f(x)]$, where $[\cdot]$ is the greatest integer functin, and $f^{-1}(\mathrm{x})$ are both continuous.
  • Correct answer
    C
    $\tan [f(x)]$, where $[\cdot]$ is the greatest integer function, and $\frac{1}{f(\mathrm{x})}$ are both discontinuous.
  • D
    $\tan [f(x)]$, where $[\cdot]$ is the greatest integer function, is discontinuous but $\frac{1}{f(\mathrm{x})}$ is continuous.

Solution

$\begin{aligned} & \mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{2}-1, \quad[0, \pi] \\ & \tan \cdot \mathrm{f}(\mathrm{x})=\tan \left(\frac{\mathrm{x}}{2}-1\right)\end{aligned}$ $\frac{1}{f(x)}=\frac{1}{\frac{x}{2}-1}$ is discontinuous at $x=2$ $\tan . \mathrm{f}(\mathrm{x}) \text { is discontinuous for } \mathrm{x}=2 \text { in }[0, \pi]$

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

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