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Question: Answered & Verified by Expert
If $f(\mathrm{x})=\frac{\mathrm{x}}{2}-1$, then on the interval $[0, \pi]$ which one of the following is correct?
MathematicsContinuity and DifferentiabilityNDANDA 2017 (Phase 2)
Options:
  • A $\tan [f(\mathrm{x})]$, where $[\cdot]$ is the greatest integer function, and $\frac{1}{f(\mathrm{x})}$ are both continuous.
  • B $\tan [f(\mathrm{x})]$, where $[\cdot]$ is the greatest integer functin, and $f^{-1}(\mathrm{x})$ are both continuous.
  • C $\tan [f(\mathrm{x})]$, where $[\cdot]$ is the greatest integer function, and $\frac{1}{f(\mathrm{x})}$ are both discontinuous.
  • D tan $[f(\mathrm{x})]$, where $[\cdot]$ is the greatest integer function, is discontinuous but $\frac{1}{f(\mathrm{x})}$ is continuous.
Solution:
1107 Upvotes Verified Answer
The correct answer is: $\tan [f(\mathrm{x})]$, where $[\cdot]$ is the greatest integer function, and $\frac{1}{f(\mathrm{x})}$ are both discontinuous.
$\begin{aligned} & \mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{2}-1, \quad[0, \pi] \\ & \tan \cdot \mathrm{f}(\mathrm{x})=\tan \left(\frac{\mathrm{x}}{2}-1\right) \end{aligned}$
$\frac{1}{\mathrm{f}(\mathrm{x})}=\frac{1}{\frac{\mathrm{x}}{2}-1}$ is discontinuous at $\mathrm{x}=2$
tan. $\mathrm{f}(\mathrm{x})$ is discontinuous for $\mathrm{x}=2$ in $[0, \pi]$

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