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Class 12 Relations And Functions · Mcq Single Correct1 Markstypes_of_functions

Class 12 Maths · Relations And Functions · Mcq Single Correct #10

For which real values of \(p\) is the function \(f:\mathbb{R}\to\mathbb{R}\), defined by \(f(x)=px+\sin x\), bijective?

✅ Step-by-Step Solution

Final Answer: Option (B): \((-\infty,-1]\cup[1,\infty)\)
  1. If \(p\ge1\), then \(f\prime(x)=p+\cos x\ge0\), with no interval on which it is identically zero, so \(f\) is strictly increasing. If \(p\le-1\), it is strictly decreasing by the same argument. Since \(\sin x\) is bounded, these functions have opposite infinite limits at the two ends of the real line and are onto. For \(|p|<1\), the derivative takes both positive and negative values; the continuous function is not monotone and hence not injective. Thus \(p\in(-\infty,-1]\cup[1,\infty)\).

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