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Course Queries Syllabus Queries 2 years ago
Posted on 16 Aug 2022, this text provides information on Syllabus Queries related to Course Queries. Please note that while accuracy is prioritized, the data presented might not be entirely correct or up-to-date. This information is offered for general knowledge and informational purposes only, and should not be considered as a substitute for professional advice.
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Question
Suppose f:R→Rf:ℜ→ℜ is a real valued function defined on the whole real line. For each a) and b) determine if the statement is correct and justify your answer.
a) If f(x)f(x) is even then g(x)=⌊f(x)⌋g(x)=⌊f(x)⌋ even.
b) If f(x)f(x) is odd then g(x)=⌊f(x)⌋g(x)=⌊f(x)⌋ odd.
Things I know I know that the ref="https://forum.tuteehub.com/tag/floor">floor ref="https://forum.tuteehub.com/tag/function">function is the greatest integer less than the value x. f(x)=f(−x)f(x)=f(−x) if f(x)f(x) is even and the graphs are symmetrical about y=0 f(x)=−f(−x)f(x)=−f(−x) if f(x)f(x) is odd
Things I know
I know that the ref="https://forum.tuteehub.com/tag/floor">floor ref="https://forum.tuteehub.com/tag/function">function is the greatest integer less than the value x.
f(x)=f(−x)f(x)=f(−x) if f(x)f(x) is even and the graphs are symmetrical about y=0
f(x)=−f(−x)f(x)=−f(−x) if f(x)f(x) is odd
I have tried to think of a way, but with the knowledge I dont know how to come to the conclusion if the statements are correct and to justify. But my knowledge is up to date with the ref="https://forum.tuteehub.com/tag/school">school syllabus (what we have been taught already). How do I do this?
b) is incorrect. The easiest counter example is taking f(x)=xf(x)=x. Booldy's counter example is correct too because if f(x)=sinxf(x)=sinx then g(x)=⌊sin(−x)⌋=⌊−sinx⌋g(x)=⌊sin(−x)⌋=⌊−sinx⌋ but nothing ensures that ⌊−sinx⌋=−⌊sinx⌋⌊−sinx⌋=−⌊sinx⌋. Take x=π4x=π4 to be convinced.
For a) just use the definition you have. Let x∈Rx∈R.ff is even so f(−x)=f(x)f(−x)=f(x). Now use this while calculating g(−x)g(−x).
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