Course Queries Syllabus Queries 2 years ago
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b) is incorrect. The easiest counter example is taking f(x)=x. Booldy's counter example is correct too because if f(x)=sinx then g(x)=⌊sin(−x)⌋=⌊−sinx⌋ but nothing ensures that ⌊−sinx⌋=−⌊sinx⌋. Take x=π4 to be convinced.
For a) just use the definition you have. Let x∈R.f is even so f(−x)=f(x). Now use this while calculating g(−x).
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Best Answer
2 years ago
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.
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?